213 


UC-NRLF 


3  uncLer  signed,  on^foeficulf-Qj 

+,  begs  leave  to   request 


the    a,Goept'CL7iGe    of    the 


Director  of  the  Ciuoinnati  Observatory. 


ERRATA. 


PAGE 

LINE 

FOR 

/            READ 

7 

35 

third 

thread 

8 

16 

than  demanded 

'than  is  demanded 

13 
13 

19 
33 

pure 

j  mere 

16 

4 

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j  elsewhere 

20 

36 

its  principal  step 

*  in  its  principal  steps 

22     33 
23 

,34 

48 

j  thereby  measured 
{  labor  proves  itself 

here 

f  increase  of  labor  thereby 
^  j  produced  proves 

herewith 

24 

24 

has 

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25 

9 

a           (in  1st  column)     a 

27 

2 

3640    ("     "         " 

)»  3540 

28 

8 

219      (  •'  7th 

)v210 

28 

29 

13720  (  "  5th       " 

)v'  1272(» 

28 

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345      (  "  7th       " 

)  v344 

33 

3 

clock-chronometer 

^    clock  or  chronometer 

33 

9 

right 

v'  night 

33 

12 

swerving 

^reversing 

33 
34 

53 
34 

(Omit  the  last  commas/and  close  up  the  line.) 
\/ 
near  one  thread                over  one  thread 

35 

37 

for  either  of 

^  for  the  first  three  of 

36 

25 

methods 

V    method 

37 

42 

bi+bo+Aip+Aip 

y'  b^bo+Aip+Asp 

38 

7 

Erase  and  insert: 

—  bO  cos  Z—  (To—  'L\)^sin  Z 

38     48 

,49 

sin  a  (and)  cos  a 

sin^d  (and)  cos/d 

39 

11 

J=&c. 

J     t^-'&C. 

ERRATA. 


PAGE       LINE 

FOR                                                    READ 

39               45 

quantity  ,n                  <J  quantity  n 

39        —3 

(cotg  ^  —  tg  <J)                 v  (cotg  <p  —  tg  rf  cotg2^) 

40           1^' 

v=  tg  <p  cotg  rf                 v  v=tg  G>  cotg  6l 

40          47 

2  cotg2                            V  2  cotg2d> 

41     29,  39 

n2  ±  C2c                          y'  u2  qz  C2c 

42          12 

factors                            v  functions 

42          14 

Wrangel                        v/  Westphal 

GENERAL  STAR  CATALOGUE. 

STAR    NO. 

COLUMN                     FOR                                             READ 

4,    29,    53 

3               Hydrae                            Hydri 

59 

5              20m                             ^  29m 

92 

3               Geminorum                v  E  Geminorum 

109 

p  Cancri                       v  6  Cancri 

112 

3               *  (15)  Argus                ».  15  (i)  Argus 

154 

5              27s                                  25* 

170 

a1  GruciN                           a  Crucis 

170 

7               v)-\                                  22\^ 

171 

3               b  Corvi                            «5  Corvi 

199 

3               ?  Bootis                        {/6>  Bootis 

200 

7              30/x                             */36/x 

219 

3               A2  Ursae  Minoris        ^  72  Ursae  Minor 

251 

7Q10                                            y^-LOTO 
Gi                                            '    -rol 

/c/  t^JLt  fist  -  /Y  4f-  '< 

/ 

THE 


PORTABLE  TRANSIT  INSTRUMENT 


VERTICAL  OF  THE  POLE  STAR, 


TRANSLATED 


FROM  THE  ORIGINAL  MEMOIR  OF  WM.  DOLLEN, 


BY 


CLEVELAND  ABBE, 

DIRECTOR    OF   THE   CINCINNATI   OBSERVATORY. 


WASHINGTON. 

GOVERNMENT  PRINTING  OFFICE. 
1870. 


4  THE    PORTABLE    TRANSIT    INSTRUMENT    IN 

in  and  of  itself  be  sufficiently  trustworthy,  we  shall  often  do  well,  for  the 
sake  of  observing  Polaris,  to  renounce  the  other  advantages  that  accom 
pany  an  observation  made  as  near  as  possible  to  the  meridian. 

This  will,  however,  be  absolutely  necessary,  if  perhaps,  as  only  too 
easily  happens  with  the  traveler,  the  mounting  is  by  no  means  solid  enough 
to  sufficiently  assure  the  invariability  of  the  instrument  during  the  entire 
interval  necessary  to  a  complete  time  determination  in  the  meridian.  In 
fact,  under  such"  circumstances  no  other  method  remains  than  to  limit 
the  duration  of  the  observations  to  the  shortest  possible  time,  and  to 
effect  this,  therefore,  it  is  necessary  to  not  wait  until  the  proper  stars, 
especially  those  situated  near  the  pole  and  necessary  for  orientation,  have 
reached  the  meridian,  but  to  leave  the  meridian  and  to  seek  the  most 
appropriate  of  them  all,  Polaris  itself,  wherever  it  may  be  in.  its  diurnal 
circle.  The  dexterity  of  the  observer  will  then  show  itself  in  the  short 
ness  of  the  interval  which  elapses  after  or  even  before  the  transit  of  Po 
laris  over  one  of  the  threads  of  the  reticule  and  the  observation  of  the 
transit  of  the  time  star  proper,  (of  course  on  all  or  at  least  as  many 
threads  as  possible;)  and  we  know  from  much  experience  that  with  a 
little  practice  this  interval  need  be  only  a  few  minutes.  The  observa 
tion  in  the  first  position  is  completed  by  a  careful  determination  of  the 
inclination  of  the  horizontal  axis,  or,  as  is  very  desirable,  by  two  determi 
nations  inclosing  the  observations  of  the  stars,  and  whose  agreement 
bears  testimony  to  the  actual  invariability  of  the  instrument.  It  is,  how 
ever,  important  immediately  to  execute  a  similar  series  of  observations 
in  the  other  position  of  the  instrument,  in  order  to  free  the  result  from 
the  influence  of  the  error  of  collimation  and  the  difference  in  diameters 
of  the  pivots,  without  being  under  the  necessity  of  taking  these  quanti 
ties  from  other  sources. 

It  is  easy,  and  will  certainly  also  be  intended  in  these  observations,  to 
observe  Polaris  both  times  upon  the  same  thread,  by  preference  the  mid 
dle  thread,  and  the  succeeding  computation  will,  in  fact,  be  thereby 
somewhat  simplified.  One  easily  sees,  however,  that  so  far  as  concerns 
the  true  object  of  the  observation,  this  is  quite  without  serious  import 
ance,  while  in  the  execution  of  the  observation  a  great  relief  may  result 
from  not  being  bound  to  any  such  condition.  This  stands  in  connection, 
however,  with  certain  imperfections  of  the  transit  instruments  in  their 
present  construction,  to  which  the  attention  of  the  observer  deserves  to 
be  especially  called. 

3.  The  instrument  to  which  the  following  remarks  are  especially  ap 
plicable  is  theErtel  portable  transit  instrument,  although  they  certainly 
have  also  a  more  general  importance.  This  instrument,  in  different 
forms,  but  all  of  nearly  identical  construction,  has  attained  an  extensive 
distribution,  and  may,  indeed,  through  the  descriptions  given  in  words 
and  drawings  in  different  authorities,  be  considered  as  generally  known. 
It  differs  chiefly  from  the  formerly  much-used  Trough  ton's  transit  in  the 
broken  telescope  and  the  possibility  of  the  motion  about  the  vertical 
axis.  The  former  secures  a  marked  facility  in  the  observation,  especially 
in  the  neighborhood  of  the  zeriith,.and  by  reason  of  the  shortening  of  the 
supports  of  the  horizontal  axis  conduces  much  to  give  a  greater  rigidity 
to  the  entire  instrument.  The  other  change  had  certainly  as  immediate 
object  to  facilitate  the  exact  adjustment  in  any  azimuth,  especially  in 
the  prime  vertical,  but  the  slow-motion  screw  serving  thereto  and  the 
careful  divisions  of  the  horizontal  circle,  for  reading  which  four  verniers 
are  provided,  as  well  as  the  circumstance  that  by  the  pressure  of  a  sup 
porting  spring  the  motion  of  the  limb,  in  respect  to  the  alidade,  can  be 
facilitated  at  will,  give  reason  to  suspect  that  the  additional  design  was 


THE    VERTICAL    OF    THE    POLE    STAR.  5 

entertained  of  rendering  possible  the  exact  measuring  of  horizontal 
angles. 

But  since  any  such  intention  is,  through  the  absence  of  the  assurance 
(or  watch)  telescope,  only  to  a  limited  degree  attainable,  the  greater  mo 
bility  in  azimuth  therefore  directly  and  very  seriously  endangers  the 
excellence  of  the  instrument  as  a  transit.  The  clamp  that  ought  to  hold 
fast  the  two  circles,  with^'efereiice  to  each  other,  and  thereby,  therefore, 
the  moveable  upper,  with  reference  to  the  immoveable  lower  portion, 
performs  this  service  very  imperfectly,  not  only  because  it  operates  only 
on  one  point,  and  that  a  point  on  the  circumference,  but  also  because  of 
the  slow  motion  that  is  combined  with  it.  Very  soon,  therefore,  it  was 
that  two  simple  clamping  screws,  distant  180°  from  each  other,  were  ap 
plied,  wfiich  are  to  be  tightened  as  soon  as'  the  instrument  is  brought 
into  the  proper  azimuth ;  and  these  are  quite  well  adapted  to  greatly 
increase  the  security  of  the  mounting. 

On  the  other  hand,  not  only  is  the  exact  adjustment  in  a  known  azi 
muth,  as  given  by  the  readings  of  the  circle,  made  almost  impossible  by 
reason  of  these  clamping  screws,  since  their  greater  or  less  tightness  is 
accompanied  by  sensible  flexures  and  corresponding  derangements  in 
the  azimuth,  but,  furthermore,  directly  through  these  flexures  is  the  ful 
fillment  of  the  other  condition,  that  the  inclination  of  the  horizontal  axis 
shall  be  always  the  least  possible,  made  much  more  difficult.  These 
clamp  screws  can,  moreover,  if  they  are  not  applied  to  the  proper  points, 
give  rise  to  a  further  danger,  against  which  one  must  be  forewarned. 
Evidently  they  should  only  be  placed  at  the  points  where  the  supports 
of  the  horizontal  axis  are.  In  the  earlier  instruments  they  are,  indeed, 
always  found  at  these  points,  and  only  recently  they  have  been  placed 
90°  distant  therefrom,  probably  for  the  sake  of  greater  convenience  in 
the  manipulation;  this  is,  however,  precisely  where  they  least  of  all  re 
alize  their  object;  for  if  the  previously-mentioned  supporting  spring  be 
even  very  slightly  compressed,  then,  notwithstanding  the  tightening  of 
the  misplaced  clamp  screws,  the  supports  of  the  Y's  will  remain  in  a  more 
or  less  unstable  position,  and  thereby  is  generated  the  danger  that  the 
inclination  of  the  axis  may  change  when  the  level  is  set  upon  it. 

We  have  persuaded  ourselves,  by  direct  trials,  that  the  danger  is  not 
a  fancied  one,  but  that  in  the  above  way  very  sensible  errors  can  arise. 
These  become  very  apparent  if  on  such  an  instrument  with  the  spring 
compressed  a  series  of  levelingsin  alternate  positions  is  made,  as  though 
for  the  determination  of  the  difference  of  the  pivot  diameters.  In  case 
the  necessary  care  in  the  reversions  has  been  taken,  we  may  certainly 
receive  very  accordant  but  entirely  deceptive  values ;  for,  the  influence 
of  the  unequal  pivot  diameters  will  have  entirely  disappeared,  in  com 
parison  with  that  of  the  unequal  weights  of  the  two  sides  of  the  instru 
ment.  Therefore  it  is  indispensable  that  in  using  such  an  instrument 
the  spring  should  be  perfectly  slack,  in  order  that  before  the  tightening 
of  the  clamp  screws  the  upper  portion  may  rest  entirely  around  with  its 
Avhole  weight,  even  though  thereby  the  slow  motion  in  one  direction  will, 
to  a  great  extent,  refuse  to  work.  On  no  account,  however,  should  the 
circumspect  observer  neglect  to  make  a  special  investigation  with  the 
object  of  determining  whether  the  application  of  the  level  itself  does  not 
still  alter  the  inclination  of  the  axis. 

4.  What  precedes  will  quite  suffice  to  show  that  some  experience  is 
necessary  in  order  successfully  to  conduct  the  series  of  observations  in 
that  rapid  succession  which  constitutes  the  importance  of  this  method, 
as  also  that  it  affords  a  very  important  relief  to  be  allowed  to  observe  the 
Pole  Star  on  any  thread,  but  especially  upon  different  threads,  in  the 


6  THE  PORTABLE  TRANSIT  INSTRUMENT  IN 

two  positions.  This  latter  finds  its  full  importance  if  it  is  thereby  made 
possible  to  preserve  the  former  azimuth  alter  the  reversion.  Independ 
ent  of  the  convenience  which  thus  results  to  the  observer,  since  he  needs 
only  once  to  clamp  his  instrument  and  adjust  it  for  the  inclination  of  the 
axis,  the  accuracy  of  the  observation  itself  will  thereby  undoubtedly  be 
increased,  since  it  is  well  known  that  for  some  time  alter  every  change 
in  the  position  of  the  instrument  there  remains  the  danger  of  a  reaction 
ary  change;  and  finally  one  finds  a  very  acceptable  control  for  the  entire 
operation  in  that  for  each  position  the  same  azimuth  should  result,  al 
though  so  far  as  the  determination  of  time  is  concerned  it  is  only  neces 
sary  that  the  position  of  the  instrument  should  not  have  changed  during 
the  observation  in  each  position  by  itself. 

This,  however,  it  must  be  confessed,  at  least  for  our  present  instru 
ments,  assumes  that  the  Pole  Star  has  a  definite  motion  in  azimuth. 
Therefore,  in  the  immediate  neighborhood  of  its  elongation  this  advan 
tage  must  be  renounced,  and  we  are  reduced  to  the  necessity  of  adjusting 
anew  the  instrument  in  each  position  upon  the  almost  motionless  star. 
For,  that  Polaris  becomes  in  general  in  this  portion  of  its  daily  orbit 
less  proper  for  our  object,  and  therefore  may,  or  even  must,  be  replaced 
by  another  star  in  the  neighborhood  of  the  pole,  perhaps  by  3  Ursae  Mi- 
noris,  is,  as  it  seems,  a  wide-spread  yet  manifest  error.  This  has  proba- 
ably  arisen,  first,  from  the  fact  that  the  determination  of  the  moment 
when  the  star  bisects  the  thread  will  truly  be  then  very  inaccurate,  and 
that,  secondly,  the  reduction  of  such  a  transit  from  the  thread  to  the 
great  circle  will  always  be  large,  and,  under  some  circumstances,  infinite. 

Now,  as  concerns  the  first  point,  we  shall,  on  consideration,  recognize 
that  therein  not  only  no  disadvantage  exists,  but  there  even  results  an 
advantage  to  the  actual  object  of  the  observation,  since  precisely  then 
the  orientation  of  the  instrument  is  achieved  as  accurately  as  the  opti 
cal  force  of  the  telescope  will  at  all  allow;  and  in  reference  to  the  sec 
ond  point  it  is  to  be  remarked,  that  this  is  a  difficulty  existing  only  in  a 
certain  method  of  computation,  and  therein  is  a  proof  that  this  method 
is  not  the  proper  one.  No,  if  we  are  to  leave  the  meridian  at  all  it  must 
be  only  for  the  sake  of  the  advantage  which  the  Pole  Star  offers;  there 
is  really  no  sufficient  reason  for  choosing  any  other  star  for  the  orienta 
tion  than  Polaris. 

At  the  same  time,  however,  it  is  not  to  be  denied  that  the  observation 
in  the  neighborhood  of  the  elongation  has  its  special  difficulties.  We 
must,  that  is  to  say,  not  Avait  until  the  star,  by  its  motion,  is  brought 
upon  the  thread,  but  must  adjust  the  instrument  directly  upon  the  star, 
Avhile,  as  above  remarked,  the  azimuth  will  again  be  changed  by  the 
fastening  of  the  clamping  screw.  With  some  practice,  however,  this 
difficulty  is  oveicome,  in  that  the  clamp  screw  itself  can,  up  to  a  certain 
point,  replace  the  slow  motion;  but  it  is  thereby  always  necessary  care 
fully  to  watch  that  the  simultaneously  changing  inclination  of  the  hori 
zontal  axis  remains  small  enough  to  allow  of  its  accurate  measurement. 
Furthermore,  the  correction  of  the  inclination  by  means  of  the  foot 
screws  offers  a  means  of  changing  a  little  the  position  of  the  star  with 
reference  to  the  threads,  which  can  be  occasionally  very  serviceable  to 
the  expert  observer  for  the  attainment  of  his  object,  especially  if  the  con 
struction  of  the  instruments  allows  the  level  to  remain  upon  the  axis 
during  the  observation.  Still  another  means  of  facilitating  the  observa 
tion  at  the  elongation  consists  in  the  use  of  a  somewhat  inclined  thread ; 
we  have,  however,  on  this  point  no  experience  as  to  how  advantageous 
this  would  prove  itself  in  practice. 

5.  I  allow  myself,  finally,  two  further  remarks  concerning  the  arrange- 


THE    VERTICAL    OF    THE    POLE    STAR.  7 

merit  of  the  observations,  regard  to  which  can  be  of  practical  importance 
under  certain  conditions.  If  we  relinquish  the  maintenance  of  the  same 
azimuth  in  both  positions  of  the  instrument,  it  is  then  occasionally  pos 
sible  to  secure  the  other,  in  some  circumstances  not  unimportant,  ad 
vantage  of  being  able  to  observe  the  same  time  star  in  both  positions. 
This  is  especially  important  if  no  other  time  star  of  sufficient  brilliancy 
is  at  hand ;  so  that  one  must,  without  this  resort,  content  one's  self  with 
observations  in  one  position.  The  application  of  such  a  method  fails  on 
the  approach  of  the  time  star  to  the  zenith,  on  account  of  its  increased 
azimuthal  change.  Neglecting  the  error  of  collimation  and  the  advance 
of  the  Pole  Star  in  azimuth,  both  which  can  affect  the  circumstances  as 
well  favorably  as  unfavorably,  then  in  the  latitude  of  60°  a  change  in 
azimuth  of  nearly  1°  corresponds  to  a  thread  interval  of  15' ;  that  is  to 
say,  after  the  reversion  the  instrument  must  be  changed  in  azimuth  by 
this  quantity  in  order  still  to  be  able  to  observe  the  Pole  Star  on  the 
same  thread ;  and  it  is  easy  to  decide  what  change  in  the  hour  angle  of 
the  time  stars  corresponds  to  this  change  in  azimuth.  For  a  Bootis,  for 
instance,  it  amounts  to  nearly  three  minutes  of  time,  and  it  depends  only 
upon  the  expertness  of  the  observer  and  the  arrangement  of  the  instru 
ment  whether  this  interval  suffices  for  the  reversion  and  adjustment  in 
azimuth. 

The  other  remark  refers  to  the  difficulty  known  to  all  observers  of  ob 
serving  the  transit  of  Polaris  if  the  star  is  very  faint.  Often  one  catches 
the  star  very  fairly  at  some  distance  from  the  thread,  while  on  approach 
ing  it  it  completely  disappears.  The  observation  is  then  usually  made 
by  noting  the  disappearance  of  the  star  on  one  side  of  the  thread  and  its 
reappearance  on  the  other,  and  the  mean  of  both  moments  is  considered 
as  the  time  of  transit.  Of  course,  however,  such  an  observation  is  far 
less  accurate  than  if  the  star  remains  visible  upon  the  thread  itself,  and 
it  will  therefore  be  the  less  available  the  longer  the  interval  between  the 
disappearance  and  reappearance.  Now,  we  have  in  such  cases,  with  the 
best  results,  replaced  the  observations  upon  the  thread  by  the  observa 
tion  in  the  middle  between  two  threads,  and  find  that  with  an  appro- 
J  priate  thicd.  interval  there  results  thereby  scarcely  an  appreciably  di 
minished  accuracy  of  observation. 

6.  If  we  review  the  previous  remarks  it  results  that  for  the  determi 
nation  of  time  by  means  of  a  portable  transit  instrument,  the  method  of 
observation  here  considered  is  only  in  very  special  cases  inferior  to  the 
observation  in  the  meridian ;  on  the  other  hand,  in  other  circumstances 
which  far  more  frequently  present  themselves,  it  will,  by  this  method 
only,  be  possible  to  realize  any  determination  at  all  ;  and  therefore  the 
advice  seems  to  be  authorized,  that  we  do  not  shun  the  labor  of  putting 
ourselves  by  means  of  the  necessary  practice,  to  a  certain  extent  at  least, 
in  possession  of  all  the  means  which  the  instrument  can  offer  to  the  ob 
server  to  whom  it  is  intrusted. 

The  judgment  must,  however,  be  quite  different  if  it  should  be  possi 
ble  to  give  to  the  instrument  an  arrangement  by  which  the  various  diffi 
culties,  above  mentioned,  are  still  further  removed  from  the  observation. 
And  this  is  now  in  reality  very  completely  attainable,  and  that  through 
changes  which  have  already  proved  themselves  as  individually  quite  ap 
plicable,  although  their  aggregate  combination  has  still  first  to  stand  the 
proof  of  actual  experience.  Now,  for  our  purpose,  the  most  important 
addition  to  our  present  transit  instruments  seems  to  me  to  be  the  intro 
duction  of  a  movable  thread,  whose  position,  with  respect  to  the  fixed 
threads,  will  be  any  time  recognized  by  means  of  the  micrometer  screw 
that  carries  it.  This  change,  since  it  only  affects  the  ocular,  is  also  com- 


8  THE    PORTABLE    TRANSIT    INSTRUMENT    IN 

paratively  easy  to  effect  in  the  existing  instruments.  The  decisive 
success  which  is  actually  attained  with  the  large  fixed  instruments  in 
different  places  shows  that  by  proper  care  and  circumspection  in  the  ap 
plication  and  use  of  such  a  micrometer,  an  accuracy  and  reliability  can 
be  attained  that  quite  suffices  for  our  purpose.  But  the  importance  due 
to  the  movable  thread  in  the  portable  instrument  will  be  only  A~ery  im 
perfectly  estimated  by  the  advantage  that  has  resulted  from  its  applica 
tion  to  the  large  fixed  instruments  5  for,  important  as  may  be,  for  these 
latter  instruments,  the  facility  of  observation  thereby  secured,  the  method 
of  observing  remains  unchanged,  since,  if  we  will  free  ourselves  as  com 
pletely  as  possible  from  the  assumption  of  the  invariability  of  the  mount 
ing,  then  a  meridian  mark  is  indispensable.  On  the  other  hand,  the 
movable  thread,  to  a  certain  degree,  provides  a  meridian  mark  for  the 
portable  instrument.  This  is  the  Polar  Star  itself,  that  can  now  on  any 
point  of  its  daily  orbit  be  with  equal  convenience  and  reliance  observed 
without  a  greater  consumption  of  time  than  demanded  to  direct  the  tel 
escope  and  point  the  micrometer  thread  upon  the  star,  precisely  as  if  a 
mark  were  observed. 

Together,  however,  with  this  important  auxiliary  to  the  achievement 
of  an  accurate  orientation  of  the  instrument  at  any  instant,  provided  al 
ways  the  optical  force  of  the  instrument  allows  the  Pole  Star  to  be  actu 
ally  perceived,  are  still  two  further  changes  in  the  instrument  extremely 
desirable,  if  we  would  impart  to  the  entire  determination  of  time  the 
greatest  degree  of  security  and  convenience.  That  is  to  say,  it  is  to  this 
end  necessary,  first,  that  not  only  during  the  observation  itself  but  also 
during  the  reversion  of  the  instrument  the  level  should  remain,  upon  the 
axis ;  and  second,  that  this  reversion  should  be  accomplished  much  more 
rapidly,  and  especially  more  safely — that  is  to  say,  without  danger  of  any 
other  change  than  is  attainable  in  the  reversion  with  the  unaided  hand. 

The  possibility  of  a  perfect  execution  of  these  arrangements,  even  in 
smaller  instruments,  is  circumstantially  shown  by  experience;  but 
truly  they  affect  the  construction  of  the  entire  instrument  so  vitally  that 
they  must,  in  planning  it,  necessarily  be  kept  in  view.  The  increased 
labor  and  care  to  be  expended  in  the  construction  of  such  an  instrument, 
and  the  increased  costliness  dependent  thereon,  should  not,  it  seems, 
come  into  consideration  in  contrast  with  the  important  increase  in  the 
value  of  the  observations  resulting  therefrom.  Such  expenditure  will 
be,  even  in  a  brief  use,  richly  repaid  by  the  saving  in  labor  at  every  ap 
plication,  without  its  being  necessary  especially  to  mention  the  further 
advantage  that  is  found  in  the  diminution  of  the  danger  of  seeing  the 
series  of  observations  that  has  been  commenced  interrupted  before  its 
completion. 

How  important  in  reality  is  the  diminution  of  the  duration  of  the  ob 
servations  which  results  from  our  method  with  instruments  specially 
constructed  therefor,  will  be  recognized  if  one  once  considers  the  course 
pursued  in  a  complete  determination  of  time.  As  to  what  concerns  the 
preparation  for  the  observation,  that  consists  mostly  in  clamping  the  in 
strument  with  sufficiently  small  inclination  of  the  axis  in  such  an  azi 
muth  that  the  Pole  Star  appears  approximately  in  the  middle  of  the 
reticule.  As  soon,  then,  as  the  observation  of  the  transit  of  a  proper 
time  star  is  finished,  the  level  and  Pole  Star  are  read  off;  a  re  version  of 
the  level  upon  the  axis  is  not  necessary  if  we  at  once  reverse  the  instru 
ment  itself  and  thereby  intentionally  disturb  neither  inclination  nor  azi 
muth.  A  second  time  star  in  this  other  position  of  the  instrument  with 
the  thereto  belonging  readings  of  level  and  Pole  Star,  completes  the  de 
termination  of  time. 


THE    VERTICAL    OF    THE    POLE    STAR. 

It  is  seen  that  the  duration  of  the  whole  depends  mostly  only  upon 
ho\v  quickly  the  proper  time  stars  follow  each  other  in  the  heavens,  and 
will,  therefore,  in  general,  not  need  to  exceed  a  very  small  number  of 
minutes.  Finally,  however,  it  is,  after  all,  not  this  decided  saving-  of 
labor,  important  as  it  in  itself  may  be,  that  gives  the  desired  contraction 
of  the  duration  of  the  observations  its  proper  value,  but  this  rests  rather 
upon  the  increase  of  the  accuracy  that  arises  therefrom.  I  estimate  this 
to  be  very  considerable.  But  to  enable  any  such  opinion  to  claim  a 
universal  recognition,  it  must  be  based  upon  unalterable  numerical 
values,  which  can  only  be  obtained  by  actual  application  continued  for 
a  long  time  and  under  the  greatest  variety  of  exterior  circumstances. 
Such  numbers  are  at  this  moment  not  at  my  command,  and  I  therefore 
content  myself  with  illustrating,  by  the  following  considerations,  the 
opinion  that  I  have  formed  after  many  years'  experience  in  the  matter. 

One's  first  thought  may  be  to  increase  the  accuracy  of  the  time  deter 
mination,  if  not  to  any  at  least  to  a  considerably  higher  degree,  by  the 
repetition  of  the  observations.  If  we  estimate  about  fifteen  minutes  for 
the  duration  of  a  complete  determination,  which,  according  to  what 
precedes,  is  considerably  more  than  what  will  generally  be  necessary, 
then,  in  an  hour,  which  in  general  certainly  would  not  suffice  for  a  com 
plete  time  determination  in  the  meridian,  this  may  be  four  times 
repeated,  whereby  the  probable  error  of  the  final  clock  correction,  in  so 
far  as  it  depends  upon  the  astronomical  observation,  would  be  reduced 
to  one-half.  Xow  I  am  of  the  opinion  that  this  would  be  practically  of 
no  use  whatever.  That  is  to  say,  I  think  that  already  the  single  deter 
mination  possesses  such  a  degree  of  accuracy  that,  independent  of  error 
in  the  star  places,  it  cannot  be  further  increased  by  multiplication  of 
observations ;  even  the  circumstance  that  a  uniformity  in  the  chronom 
eter  rate  must  then  be  assumed  for  so  much  longer  an  interval,  suffices 
alone  to  annul  the  advantage  of  reducing  by  one-half  the  error  of  obser 
vation.  I  mean,  in  all  seriousness,  that  the  probable  error  of  such  a 
time  determination,  so  far  as  the  observation  has  an  influence  on  it, 
depends  principally  i\pon  nought  else  than  the  accuracy  of  the  observed 
transits  over  the  single  threads,  which  evidently  is  the  limit  of  accuracy 
that  any  method  whatever  can  afford.  And  should  this,  as  I  confidently 
hope,  be  confirmed  by  unprejudiced  examination,  then  will  the  expres 
sion  not  seem  too  venturesome  that  in  the  construction  of  a  portable 
transit  instrument,  of  moderate  dimensions,  according  to  the  principles 
above  laid  down,  and  in  artistic  perfection,  such  as  we  shall  in  a  short 
time  receive  from  the  hands  of  Brauer,  a  new  epoch  begins  in  the  solu 
tion  of  that  very  important  problem,  the  absolute  determination  of  time. 

7.  Finally,  I  cannot  refrain  from  noticing  Avith  a  few  words  more  par 
ticularly  the  especial  occasion  which  has  been  before  alluded  to,  and 
which  renders  it  extremely  desirable  that  precisely  now  the  great  ad- 

V£W1  i"tl  O*4-*t2      f\T     "f"li«-i      1  vi  oi"l  k  /"wl       /"v~f*     /I  ^'rf-nv»i'v*  iTi  1 11  rv      'i-in^n       \\  m*s\       ^/-ii»  r»4  *•!  s^-rn-\  /I      ^.T-*  ^.-.-.1  /I 


vantages  of  the  method  of  determining  time  here  considered  should 


given  a  start  to  which  is  a  fit  conclusion  of  the  labors  by  which  W. 
Struve  has  so  ably  promoted  geodesy  during  his  whole  life. 

Already  is  the  determination  of  the  astronomical  differences  of  longi 
tudes  along  the  entire  arc  appointed  for  the  next  two  years.  The  trans 
mission  of  time  will  everywhere,  when  possible,  be  made  by  means  of 
the  electric  telegraph.  Since,  however,  at  present  the  telegraph  does 
not  directly  connect  the  different  points  of  the  arc,  the  desired  differ- 


10  THE    PORTABLE    TRANSIT    INSTRUMENT    IN 

ences  of  longitude  Avill  be  obtained  by  means  of  the  central  telegraph 
stations  of  the  different  States.  Tims  it  will  be  possible  that  all  the 
time  determinations  upon  the  entire  arc  shall  be  made  by  the  same 
observer  with  the  same  instrument  ;  and  furthermore,  it  is  provided 
that  this  shall  be  entirely  independently  carried  out  by  the  two  different 
observers,  so  tbat  for  each  individual  difference  of  longitude  the  neces 
sary  clock  correction  shall  depend  upon  the  quite  independent  time 
determination  of  each  observer.  It  is,  however,  equally  designed  that 
each  observer  should  be  assigned  to  one  and  the  same  instrument. 
Now,  this  instrument  will  indeed,  in  order  to  facilitate  the  labor,  be 
provided  with  a  level  that  does  not  need  to  be  removed  during  the 
observation,  and  with  a  special  apparatus  for  reversion.  And  still  I 
hold  decidedly  to  the  opinion  that,  independent  of  other  difficulties,  the 
fulfillment  of  the  programme  for  any  night's  observation,  under  the  con 
ditions  connected  with  a  complete  time  determination  in  the  meridian, 
will  be  possible  only  under  very  specially  favorable,  and  therefore  seldom 
occurring,  exterior  conditions,  especially  if  we,  as  certainly  must  be 
considered  necessary,  will  not  relinquish  the  condition  that  the  trans 
mission  of  time  be  each  time,  and  as  closely  as  possible,  included  by 
time  determinations  at  each  place. 

Now,  under  these  circumstances,  the  method  here  recommended 
seems  to  be  an  especially  fortunate  expedient.  Assuming  our  reticule 
to  consist  of  seven  threads,  then  the  two  stars  twice  observed  by  each 
observer  give  a  sum  total  of  fifty-six  individual  thread  transits,  and  thus  a 
time  determination  that  is  quite  comparable  with  a  time  transmission 
by  means  of  as  many  telegraphic  signals.  And  the  whole  time  neces 
sary  for  this  operation  need  be  scarcely  an  hour.  It  seems  useless  to 
attempt  to  increase  the  accuracy  of  the  determination  for  one  night  by 
repetition,  as  is  easily  possible ;  imperative,  however,  is  the  repetition 
on  different  nights,  as  is  also  prescribed  in  the  adopted  programme. 
The  final  determination  of  the  details  in  this  respect,  as  also  in  general, 
properly  remains  deferred  until  the  accurate  investigations  which  will  be 
made  in  the  immediate  future,  as  preparatory  to  the  work  of  the  next  years, 
be  completed.  For  the  present  we  are  concerned  only  with  the  acknowl 
edgment  of  a  fundamental  principle  which  I  would  thus  enounce: 

FOR  THE  ATTAINMENT  OF  A  TIME  DETERMINATION  AT  A  GIVEN  MO 
MENT,  THE  PORTABLE  TRANSIT  INSTRUMENT  WILL,  UNDER  ALL  CIR 
CUMSTANCES,  BE  BEST  MOUNTED,  NOT  IN  THE  MERIDIAN  BUT  IN  THE 
VERTICAL  OF  THE  POLE  STAR. 

The  next  months  will  bring  us  the  final  practical  proof  of  the  truth 
of  this  thesis. 

8.  When  the  question  as  to  the  practical  value  of  the  method  of  obser 
vation  here  considered  is  once  settled,  then  will  the  overcoming  of  the 
greater  labor  of  computation  be  an  object  of  comparatively  secondary 
importance.  A  nearer  consideration  shows,  however,  that  this  is  by  no' 
means  so  excessive  as  it  appears  at  the  first  view ;  and  the  relation  is 
even  reversed  in  those  cases  where,  for  the  meridian  time  determination, 
we  dare  not  be  contented  without  the  exact  solution  according  to  the 
method  of  least  squares. 

The  deduction  of  methods  of  computation  which  shall  unite  the  great 
est  convenience  of  execution  with  the  necessary  accuracy  of  the  results, 
has  been  undertaken  at  different  times  and  in  different  places,  especially 
lately  on  the  part  of  Hanseu,  in  an  exhaustive  manner,  in  an  article  in 
No.  1136,  Vol.  XLYIII,  of  the  Astronomische  Nachrichten.  Starting 
from  the  known  general  equation  for  the  transit  instrument,  Hansen 
gives  first  a  direct  exact  solution  by  purely  analytical  means,  which  is 


THE    VERTICAL    OF    THE    POLE    STAR.  11 

well  worthy  of  consideration  by  reason  of  the  art  with  which  the 
unknowns  are  found  from  the  transcendental  equations.  Since,  how 
ever,  there  can  ne>ver  be  sufficient  ground  to  make  use  of  this  exact  but, 
therefore,  more  tedious  solution,  there  are  next  deduced  approximate 
formulae  for  actual  use,  which  really  leave  scarcely  anything  more  to 
be  desired.  Although  after  this  every  further  discussion  of  the  subject 
might  seem  idle,  yet  I  venture  in  what  follows  to  present  the  somewhat 
different  method  that  1  for  years  have  followed  in  treating  this  problem, 
and  that  in  very  frequent  applications  has  always  proved  itself  to  my 
mind  thoroughly  appropriate.  The  direct  exact  solution  to  which  I 
arrive  at  first  seems  to  me  to  recommend  itself  as  very  advantageous, 
by  reason  of  the  exceeding  clearness  of  the  geometrical  considerations 
on  which  it  is  based;  afterwards,  however,  this  leads  to  approximate 
formula}  and  precepts  for  computation  which  are  so  simple  and  conve 
nient,  and  especially  so  entirely  free  from  uncertainty  in  reference  to 
the  signs  of  all  the  quantities  coming  under  consideration,  that  they 
perhaps  deserve  consideration  even  when  compared  with  Hansen's. 

9.  For  greater  generality,  and  in  conformity  with  the  above  remarks, 
I  in  the  following  development  assume  that  the  two  stars  are  not  observed 
on  the  same  thread.  After  wrhat  more  immediately  follows  will  be 
found  the  reduction  that  is  necessary  if  the  time  star  is  observed  on 
more  than  one  thread.  For  the  present  tjie  thread  on  which  the  time 
star  is  observed  is  called  the  middle  thread,  and  the  thread  for  the 
Pole  Star  is  distant  by  /from  this  5  thus,  90°  +  c  and  90°  -f  c-f/  desig 
nate  the  distance  from  the  west  end  of  the  horizontal  axis  of  the  transit 
instrument  of  the  celestial  points  determined  by  these  two  visual  lines. 

The  exact  solution  of  our  problem  consists  fundamentally  in  this,  to 
find  the  diagonal  WP  and  the  angle  WPS  of  a  spherical  quadrilateral 
WSPS'',  in  which  the  four  sides  and  the  angle  P  are  given.  Let  us 
represent  by  W  and  P  the  points  in  the  heavens  to  which  the  axes  of 
revolution  of  the  instrument  and  of  the  earth  are  directed,  and,  to  leave 
nothing  uncertain,  the  west  pole  of  the  instrument  and  the  north  pole 
of  the  equator;  S  and  S'<,  on  the  other  hand,  the  places  of  the  Time  Star 
and  of  the  Pole  Star  at  the  moment  of  observation;  we  thus  kuowr  at 
first, 

c,  WS'=9(P  + 


PS=90°  —  3,  PS/ 

Finally,  the  angle  SPS7,  which  Ave  will  designate  by  r,  would  be  simply 
the  difference  of  the  right  ascensions  of  the  two  stars,  if  both  were 
observed  at  the  same  moment.  Since,  now,  some  time  at  least  must 
elapse  between  these  two  observations,  this  difference  is  still  to  be 
changed  by  the  hour  angle  corresponding  to  this  interval;  and  in  this 
nothing  further  is  assumed  than  that  we  know  the  clock  rate  sufficiently 
well  to  make  this  change  in  hour  angle  with  an  accuracy  corresponding 
to  the  accuracy  of  the  observations.  If,  however,  the  observations  are 
arranged  as  is  proper,  then  even  without  the  movable  thread  would 
this  interval  never  be  greater  than  from  five  to  six  minutes,  and  then  a 
correction  on  account  of  the  clock  rate  would  scarcely  be  further 
demanded,  even  if  the  chronometer  kept  mean  time  instead  of  sidereal 
time;  the  estimation  of  the  moment  when  the  Pole  Star  is  on  the  thread 
is  assuredly,  with  a  portable  transit  instrument,  not  certain  to  within  a 
second,  even  in  the  neighborhood  of  the  culmination.  The  concluded 
clock  correction  belongs,  of  course,  to  the  moment  of  observation  of  the 
time  star. 

Let  a  great  circle  be  drawn  through  the  points  S7  and  S  ;  then  is  the 


12          THE  PORTABLE  TRANSIT  INSTRUMENT  IN 


exact  deduction  of  WP  and  the  angle  WPS  given  by  the  following- 
equations  : 

f  (1)      sinSS'siiiPSS'^BinPS'sinSPS' 

(I)  A  S  P  S'  :  ?  (2)      sin  S  S'  cos  P  S  S'  ==  sin  P  S  cos  P  S'  —  cos  P  S  sin  P  S'  cos  S  P  S' 
(  (3)      cos  S  S'  =  cos  P  S  cos  P  S'  -f  sin  P  S  sin  P  S'  cos  S  P  S' 

(II)  A  W  S  S':  cos  W  S  S'  =  cosWS'-cosWS  cosSS' 

sin  W  S  sin  S  S' 

t  (1)  sin  \V  1*  sin  W  P  S  =  sin  W  S  sin  W  S  P 

(III)  A  W  P  S:  J  (2)  sin  W  P  cos  W  P  S  =  sin  P  S  cos  W  S  —  cos  P  S  sin  W  S  cos  W  S  P 
t  (S)  cos  W  P  =  cos  P  S  cos  W  S  +  sin  P  S  sin  W  S  cos  W  S  P 

We  will  now  introduce  the  above-chosen  notation,  £,  <$',  e,/,  r,  into 
these  equations,  and  also  the  following: 

(7,  PS  8'=*; 


whose  immediate  object  is  merely  to  indicate  at  once  such  quantities  as, 
in  the  application,  can  never  be  other  than  small.  In  case  the  arc  S  S', 
or  the  distance  of  the  two  points  in  which  the  stars  are  observed,  is 
neither  very  near  180°  nor  near  0°,  both  which  cases  are  excluded  by 
the  nature  of  the  problem,  then  is  rt  of  the  same  order  as  c  and/,  while 
r,  w,  <r,  as  also  the  difference  d  —  <J,  are  of  the  same  order  as  P  S'. 

But,  entirely  without  reference  to  the  magnitudes  of  all  quantities,  our 
exact  formulae  are  now  the  following  : 

c  (1)  cos  d  sin  £=  cos  6'  sin  - 

(I)  A  SPS':<  (2)  cosdcosf=cosJ  sin  <5'  —  sin  dcosrf'  cos  7 
'  (3)  sin  d         —  sin  d  sin  6'  -f  cos  6  cos  6'  cos  r 


(II)  A  W  S  S':  sin  n  =  ^ 

cos  c  cos  d 

(    (1)  COSH  COS  £  =         COS6'COS(£-f  r/) 

(III)  AW  PS:  <  (2)  COSH  sin  -£=  —  cos  6  sine   +  sin  6  cose  sin 

((3)  sin  n          =      sin  6  sine    -j-cosdcoscsin^-f?;) 

Thus  far,  as  is  seen,  everything  is  entirely  independent  of  any  refer 
ence  to  the  place  of  observation  upon  the  surface  of  the  earth,  which 
reference  will  be  first  obtained  by  the  knowledge  of  the  position  of  the 
zenith,  Z,  with  respect  to  the  quadrangle  hitherto  considered.  To  this 
end  are  used  the  distances  of  the  point  Z  from  the  points  P  and  W, 
which  are  given  by  the  latitude  of  the  place  and  inclination  of  the  hori 
zontal  axis  of  the  instrument.  If  we  put,  as  usual, 

Z  P  =  9()o  —  ^  Z  W  ==  90^  —  Z>, 

where  ft  is  positive  if  the  west  end  of  the  axis  lies  above  the  horizon  : 
also,  further, 

Z  P  W  =  9Qo  —  »«,  P  Z  W  =  900  -f-  », 

there  results  : 


(IV)  A  W  P  Z-  ~~  cc  "  sec  ^ 

'  {  (2)  sin  a  =  tg  &  tg  9  +  sin  »  sec  />  sec  9 

by  means  of  which  our  problem  is"now  completely  solved.     For 

(90o  _  m)  _  (90o-_  x}=x  —  m  =  15  1 


is  the  western  hour  angle  of  the  time  star  at  its  observation  at  S,  and 
consequently  the  sidereal  time  of  this  observation  is 

a  +  t  =  S  +  «., 


THE    VERTICAL    OF    THE    POLE    STAR.  13 

if  S  signifies  the  observed  clock  time  and  u  the  correction  at  this  moment 
of  the  chronometer  to  sidereal  time.     Therefore,  finally, 


10.  Should  one  at  any  time  have  reason  to  make  use  of  these  exact 
formuhe,  I  Avould  recommend  that  they  be  applied  as  they  here  stand, 
without  thinking  of  the  introduction  of  auxiliary  angles.  These  auxil 
iary  angles  have,  for  the  computations  of  the  present  day — thanks  to 
the  increasing  dissemination  of  the  Gaussian  logarithms — lost,  to  a  great 
extent,  their  former  importance;  they  afford  a  real  relief  in  the  compu 
tation  generally,  only  Avhen  we  have  to  do,  not  with  a  single  case  but 
AAnth  many  connected  together,  to  Avhich  certain  quantities  are  common, 
as,  for  example,  often  in  the  computation  of  tables. 

On  the  other  hand,  the  experienced  computer  will  certainly  not  fail 
to  make  full  use  of  the  adA^antages  offered  by  the  smallness  of  most  of 
the  desired  quantities,  and  that,  too,  without  impairing  in  the  least  the 
perfect  exactness  of  the  result.  ftwuL^ 

Here,  hoAvever,  such  a  computation  has  always  a  p«*e  theoretical  in 
terest.  In  actual  practice  approximate  formuhe  suffice,  as  aboA'e  several 
times  remarked,  even  in  the  extremest  cases  which  can  be  likely  to  occur. 
These  approximate  formuhe  are  obtained,  if  Ave  derive  the  desired  quan 
tities  from  the  above  equations,  under  the  assumption  that  &  =  0,  c  =  0, 
/'=0,  and  thereupon  develop  the  coefficients  of  these  quantities,  Avith 
whose  assistance  we  can  at  pleasure  compute  their  influence  if  they  are 
not  equal  to  zero,  a  method  of  solution  which  will  be  necessary,  in  refer 
ence  to  c,  as  soon  as  the  error  of  collimation  is  not  assumed  as  known, 
but  must  first  be  found  from  these  same  observations  themselves. 

If  we  indicate  by  the  subscript  o  the  A^alues  resulting  under  the  assump 
tion  mentioned,  and  by  a  prefixed  J  the  correction  then  necessary,  so 
that,  for  instance,  u  =  ult  -f  J  ?/,  the  above  given  strict  formula;  giATe, 
without  trouble,  the  following : 

1  —  sin  d  1         f 

»  =  0  A;;=  —    — 7 — .  c  +  -  ,./ 

o  cos  d  cos  d 

cos  ??0  cos  XQ  =  cos  £ 

cos  n  o  sin  #o  —  sin  6  si  n$         cos  it  cos  x  .  A.T—  sin  u  s'mjc .  A  n  = —  cos  (5  .  c-{-   sin  6  cos  .f .  A  ?/ 

sin  -HO  =  cos  d  sin  f  cos  n  .  A  tt  •=      sind.c-f-    cos  d  cos  f .  A  » 

sin  w0          =  tg  (j)  tg  n{>  cos  m  .  A  m  =  sec  n  sec  (j> .  J)  -f-  sec'2  n  tg  0 .  A  n 

sin  «()          =  sec  o  sin  H()  cos  a  .  A  a  =  tg  <t> .  &  -f-  cos  n  sec  0 .  ^  n 

Hence,  with  help  of  equations  I,  there  folloAA^s: 

sin  o  cos  8'  sin  - 

tff  .r,,  =  Sill  d  tg  C  —  — •. ; 

COS  d  Sill  d'  —  Sin  d  COS  d'  COS  r  ; 

tg  n0  =  cotg  d  sin  .r0 ; 
sin  MO  =  tg  (p  cotg  o  sin  ,r{>. 
If,  therefore,  we  put 

tg  d  COtg  o'  =  /,  tg  <p  COtg  d  =  <>. ' 

then  Avill 

/  sin  r 
tg  ,?•  =  ^  - ,  sin  mn  =  <>.  sin  ^0  ; 

1  —  /  COS  r  ' 

and  finally, 


14          THE  PORTABLE  TRANSIT  INSTRUMENT  IN 

But, 

.    J  x  —  Am 

«=«„  +     1:> 

therefore, 

J  tn —  A  x  t 

15 
for  which  we  will  write, 

in  which  the  divisor  15  in  the  coefficients  B,  C,  F,  which  are  now  to  be 
immediately  developed,  need  not  be  further  considered  if  we  have  ex 
pressed  6,  c,/,  in  time.  We  remark,  first,  in  reference  to  this  develop 
ment,  that  in  the  above-given  differential  formula?  the  cosines  of  the  small 
angles,  fr,  c,/,  have  been  put  equal  to  unity,  and  the  sines  proportional 
to  the  angles  themselves,  while  in  the  case  of  the  small  quantities,  ?,  x, 
n-,  m,  «.,  this  simplification  has  not  yet  been  introduced.  This  is  proper, 
since  that  these  latter,  although  small  in  themselves,  are,  or  at  least  can 
become,  many  times  larger  than  the  former.  Of  course  b  will  naturally 
amount  only  to  a- few  seconds  of  arc,  since  larger  inclinations  cannot  be 
measured  with  the  level.  The  intelligent  observer  will  furthermore  take 
care  that  c  be  always  very  small,  but  it  is  not  at  all  meant  to  advise  to 
attempt  to  make  it  zero  every  time.  Finally,  in  the  most  extreme  cases 
f  can  certainly  be  so  large  as  15  minutes  of  arc;  the  extreme  threads  will 
certainly  never  stand  further  than  this  from  the  middle  one  ;  but  even 
then  cosine /does  not  differ  from  Radius  by  a  unit  in  the  fifth  decimal 
place.  We  will  most  easil}T  receive  an  idea  of  the  magnitudes  of  the 
other  quantities  if  we  represent  their  approximate  values  as  functions  of 
any  one,  most  conveniently  of  n.  Our  formulas  give  us  at  once : 

a  =  n  sec  y,  m  =  n  tg  cr,  z  =  n  sec '),  ,r  =  n  tg  o. 
Therefore,  if  we  put 

1  —  cos  n  =pi 

there  results,  approximately, 

1 — cos  a  =p  sec2  ^,  1 — cosw==ptg2c?,  1  —  cosr==psec2<7,  1— cos,r=^tg2  <S; 
further, 

sin  n  =  \/  '2pi    sin  x  =  tg  d  -\/  "Ip^ 
therefore, 

sin  n  sin  x  =  2p  tg  d  • 

t  and  we  will  now  easily  be  able  to  judge  how  far  these  small  fractions 
still  demand  notice.  Since  we  adopt  the  rule  to  use  no  other  star  than 
the  Pole  Star  for  orientation,  therefore  n  can  never  be  greater  than 
1°  26',  to  which  corresponds  ^  =  0.0003  ;  and  therefore  for  a  latitude  of 
60°  and  the  observation  of  zenith  stars,  none  of  the  above  quantities  be 
come  greater  than  about  0.001.  Xow  this  is  entirely  evanescent,  both 
in  respect  to  b  and  to  c,  but  not  in  respect  to  the  extreme  value  of  /, 
especially  in  case  we  would,  as  we  ought,  retain  the  hundredths  of  a  sec 
ond  of  time  in  our  results.  If  we  therefore  may  neglect  these  fractions 
at  first,  for  the  sake  of  perspicuity,  in  the  deduction  of  the  coefficients, 
B,  C,  F,  still  we  must  finally  develop  for  F  the  small  corrections  de 
manded  by  this  omission  in  order  that  this  may  always,  when  necessary, 
be  brought  into  account.  The  differential  equations,  simplified  by  the 
first  mentioned  assumption,  now  are : 

A  x  =  —  cos  o  .  c  +  sin  o  .  J  TJ 
J  n  =4-  sin  3  .  c  4-  cos  3 .  J  rt 
j  m  =  4-  sec  <f .  b  +  tg  c  .  J  n. 


THE    VERTICAL    OF    THE    POLE    STAR.  15 

From  these,  by  substitution  of  J  n  from  the  previous  equations,  there 
results, 

j  m  =  sec  <p  .  1)  4-  tg <p  sin  d.c  +  tg<p  cos  d .  J  rt  ; 
therefore, 

j  )n  —  J  x  =  sec  (p  .  b  4-  (tg  ^  sin  o  4-  cos  8) .  c  4-  (tg ^ cos  <? — sin  3)  ,Arh 

or 

j  m  —  j  #  —  sec  <p .  b  4-  sec  c^  cos  £ .  c  4-  sec  ^  sin  2 .  J  r/ , 

where  we  have  put 

(p  —  d  =  Z. 

Finally,  if  we  substitute  the  value 

j  -/}  =  (sec  d  —  tg  d) .  c  4-  sec  d  ./, 

and  unite  the  two  terms  dependent  upon  c,  the  desired  coefficients  result, 
B  =  sec  <p 

C  =  sec  tp  sec  d  [sin  z  +  cos  (d  +  z)] 
F  =  sec  (p  sec  tf  sin  z  -{-  G^>, 

Avhere  G  is  a  factor  still  to  be  deduced,  namely,  the  coefficient  of  the 
hitherto  neglected  term  in  the  value  of  F  dependent  upon  p.  In  order 
to  obtain  this  we  must  return  to  the  complete  differential  equations  and 
seek  the  value  of  the  partial  differential  coefficients  with  reference  to/, 
which,  we  will  designate  by  dx,  dn,  d  m  ;  this  will  at  the  same  time  be  a 
control  over  the  accuracy  of  the  previously-found  principal  terms  of  the 
factor  F,  independent  of  p. 
We  have 

drj  =  sec  rf, 
and  therefore 

cos  n  cos  x .  d  x  —  sin  n  sin  x  d  n  =  cos  I  sec  d  sin  3 
cos  n  d  n  =  cos  I  sec  d  cos  d 
cos  m  d  m  =  sec2  n  tg  <p  .  d  n. 

If,  by  means  of  the  second  equation,  we  eliminate  the  dn  from  the  two 
others,  we  obtain : 

cos  n  cos  x.dx  =  cos  c  sec  d  sin  3  -f-  tg  n  sin  ,r  cos  I  sec  d  cos  3 

cos  m .  d  m  =  sec3  n  tg  <p  cos  c  sec  d  cos  o, 
or  remembering  that 

cos  n  cos  x  =  cos  r , 
we  have 

d  x  =  sec  d  sin  <?  +  tg  n  sin  ^  sec  d  cos  3 

dm  =  tg  c0  sec2  n  sec  m  cos  x  sec  d  cos  £, 
and  therefore 

dm — d  x  =  F  =  sec  d  cos  d  (tg  ^  sec2  n  sec  w  cos  x  —  tg  n  sin  x  —  tg  3). 
But,  from  the  previous  remarks,  there  follows : 

sec2  n  sec  m  cos  x  =  I  4-  p  ( 2  4-  tg2  c —  tg2  o )  4- &c. 

2  » tff  3 
tg  M  sin  x =-  y-         =  2^>  tg  d  4- &c., 

therefore, 

F  =  seod  cos  3  I  tg  ?  —  tg  5  4-  #  [2  tg  ?  +  (tg2  ^  —  tg2  <5)  tg  —  2  tg  d] } 
=  sec  d  cos  o  (tg  <p  —  tg  (5)  { 1  4-  ^  [2  4-  (tg  <P  4-  tg  5)  tg  f  ] } 
=  sec  9  sec  d  sin*  { 1  +^  [2  +  (tg  <?  4-  tg  5)  tg  ?] }. 

The  portion  independent   of  p  agrees,  as  it  should,  with  that  pre- 


16  THE    PORTABLE    TRANSIT    INSTRUMENT    IN 

viously  deduced.  If  we  indicate  it  temporarily  by  F0  and  by  h,  the  factor 
of  jp,  then  will 

F  =  F0  (1  +  p  h)  =  F0  (1  +  p)*  =  F0  (sec  n)\ 

We  now  remark  that  sec  n  does  not  otherwhere  occur  in  our  computa 
tion,  and  that,  therefore,  it  would  be  convenient  to  replace  it  in  some 
way  by  seem,  which  is  found,  without  further  trouble,  in  taking  from 
the  tables  the  angle  belonging  to  m.    From  m  =  ntg  <p  there  results 
sec  n  =  (sec  m)  cot«2  0,  therefore  (sec  n)h  =  (sec  w)11  cots2  0, 

thus,  finally, 

F  =  sec  (f  sec  d  sin  z  (see  m)  k 
where 

k  =  1  +  2  cotg 2  ^  +  cotg  <p  tg  d. 

A  similar  remark  is  to  be  made  with  reference  to  the  arc  d,  the  comple 
ment  of  the  distance  between  the  two  points  of  the  heavens  S'  and  S, 
in  wrhich  the  two  stars  are  found  at  the  time  of  their  observation.  This 
arc  can  be  found,  with  all  desirable  accuracy,  from  the  equations  (I.) 
But  in  all  cases  actually  occurring  this  arc  passes,  within  a  few  seconds, 
through  the  zenith  ;  and  further,  since  it  is  only  used  in  the  computa 
tion  of  the  differential  coefficients  C  and  F.  it  is  more  than  sufficiently 
accurate  if  we  put 

00°  —  d  =  z1  -f  z,  therefore  d  -f  z  =  1)0°  —  z1, 

and  take  for  z'  the  actual  zenith  distance  of  the  Pole  Star,  counting  it 
always  positive,  while  z  remains  =  <p  —  d.  If,  as  is  for  these  observa 
tions  always  desirable,  we  have  an  ephemeris  of  the  Pole  Star,  that  is  to 
say,  a  table  of  azimuths  and  zenith  distances  with  the  argument,  sidereal 
time,  which  indeed  is  indispensable  if  the  observations  are  made  by  day 
or  twilight,  then  z1  can,  at  any  time,  be  taken  therefrom.  If  we'have 
no  such  table,  or  none  sufficiently  accurate,  we  need  only  to  read  the  ver 
tical  setting  circle  simultaneously  with  the  observation  ;  and  if  this  be 
also  done  for  the  time  star  then  the  difference  of  the  two  readings,  even 
without  any  knowledge  of  the  position  of  the  zenith,  gives  directly  the 
complement  of  the  arc  d. 

I  will  not  here  omit  to  call  attention  to  a  general  remark  that  has  very 
often  proved  itself  to  be  one  of  great  importance.  We  make  it  an  inva 
riable  rule  in  the  accurate  observation  of  either  co-ordinate  also  to  ap 
proximately  determine  the  other,  which  is  always  possible  by  means  of 
the  finding  circle.  Thus,  in  the  observations  of  transits  one  always 
reads  also  the  altitude  circle ;  in  measuring  the  zenith  distances  one 
reads  also  the  horizontal  circle;  or  rather,  we  do  not  neglect  to  record 
these  readings,  which  will  generally  be  made  in  order  to  set  the  instrument. 
Independent  of  the  thus-secured  sure  solutions  of  many  doubts  arising 
in  consequence  of  mistakes,  &c.,  these  quantities  are  always  of  import 
ance,  and  have  a  direct  application  as  soon  as  we  have  to  do  with  the 
differential  coefficients  5  and,  although  these  latter  are  not  always  used  in 
the  deductions  of  the  results  themselves,  it  is  still  very  desirable  not  to 
neglect  their  development,  when  not  entirely  too  tedious,  since  they  con 
duce  very  much  to  the  attainment  of  a  correct  judgment  as  to  the  relia 
bility  of  the  determination  obtained. 

11..  The  following  group  gives  a  complete  review  of  the  process  of  the 
computation,  according  to  the  approximate  formulae  just  developed.  Let 
there  be — 

For  the  Pole  Star.     For  the  Time  Star. 

S'  S  the  clock  time  of  observation. 

u  +  -f  u  the  correction  to  sidereal  time. 


THE    VERTICAL    OF    THE    POLE    STAR.  17 

For  the  Pole  Star.     For  the  Time  Star. 

the  apparent  right  ascension. 

o'  the  apparent  declination. 

z'  z  the  zenith  distance,  z1  is   always  posi 

tive,  z=<f>  —  3. 
We  now  form  the  quantities  : 

S/  +  ; «'  =  I)'  tg  d  COtg  8'  =  /  tg  X(>  =  r_ 

S  —a   =  T> 

15  (D'  —  D)  =  r  tg  <p  cotg  d  ==  <j.     sin  m{)  =  >j.  sin  .r0 

thus  we  have: 

where 
B==sec0  ft  the  inclination  of  the  horizontal  azis. 

C  =  sec^sl  c  the  error  of  collimation  of  the  middle 

thread. 

F  =  sec  <p   .  -.  (sec  wi-)k    /  the  distance  of  the  middle  thread  from 

sin  (z  +z]  that  ou  W}1icj1  tue  pole  Star  is  observed. 


The  inclination  b  is  positive  if  the  west  end  of  the  axis  lies  above  the 
horizon  ;  the  signs  of  c  and /are  to  be  understood  so  that  90°  -f  c  and 
90°+  c  +/ represent  the  distances  of  the  respective  points  of  the  heav 
ens  from  the  west  ends  of  the  axis.  All  three,  fr,  o,  /,  are  expressed  in 
time,  and  u  will  be  obtained  in  time. 

The  introduction  of  the  term  F/ could  evidently  be  avoided  by  always 
considering  the  thread  on  which  the  Pole  Star  is  observed  as  the  middle 
thread,  and  reducing  the  transit  of  the  time  star  to  this.  Of  course,  then, 
the  error  of  collimation  would,  in  general,  be  variable  both  in  sign  and 
also  in  magnitude  in  the  different  positions  by  the  known  value  of  the 
distance  of  the  middle  threads  adopted  in  the  two  positions.  No  advan 
tage  results,  however,  in  practice  from  this  consideration,  correct  as  it  is 
in  and  of  itself.  For,  since  in  this  case  the  collimation  error  could  have 
a  much  larger  value  than  otherwise  need  be  assumed,  therefore  the 
exact  development  which  is  now  considered  necessary  for  the  factor  F 
would  be  demanded  for  the  factor  C,  and  this  would  lead  to  expressions 
still  more  complicated  than  those  found  for  F.  At  least  I  have  not  my 
self  been  able  to  represent  the  correction  demanded  in  such  a  case  for 
the  above-given  value  of  c  more  conveniently  than  in  the  form  of  a  factor, 

(sec  w)*,  where  q  =  cosec2  y  -f  cotg  <p  tgz  ~~   i  1  —  - 

J    V       /* 

which  factor  is,  therefore,  to  be  made  use  of  if  ever  the.  collimation  error 
should  have  an  especially  large  value. 

If,  moreover,  many  time  determinations  are  to  be  made  under  the  same 
latitude  according  to  our  method,  it  is  then  certainly  most  convenient  to 
compute  the  factors  C  and  F  once  for  all,  for  the  always  moderate  num 
ber  of  stars  that  can  come  into  consideration  ;  as  we  are,  indeed,  already 
accustomed  to  do  for  the  much  simpler  factors  that  are  used  to  free  the 
transits  observed  in  the  immediate  neighborhood  of  the  meridian  from 
the  influence  of  the  different  instrumental  errors. 


18  THE    PORTABLE    TRANSIT    INSTRUMENT    IN 

12.  The  endeavor  to  further  simplify  the  deduction  of  the  clock  cor 
rection  by  any  application  of  serial  developments  has  but  little  prospect 
of.  success  ;  for,  because  of  the  magnitude  to  which  the  angles  x  and  m 
can  attain,  so  many  members  of  the  series  are  required  in  order  to  ob 
tain  accurate  results,  that  the  computation  according  to  the  exact  form 
ula  is  decidedly  more  convenient.  If,  however,  we  will  be  content  with 
an  accuracy  of  about  0s.  1,  we  may  write  : 


log          °10      =  log  (/?  Sin  r)  +  r  COS  r, 

where 

;.(/,_  i)      cot  3'(tgy  —  tgJ) 
15  sin  1"  15  sin  1" 

and 

r  =  mod  .  A  =  0.4343  cotg  8'  tg  d. 

Although  these  formulae  may  be  only  seldom  used  in  the  proper  com 
putation  of  the  observations,  they  at  least  offer  the  advantage  of  an  easy 
insight  into  the  dependence  of  the  desired  clock  corrections  upon  the 
imperfections  of  the  observations  and  the  errors  of  the  adopted  elements 
of  reduction. 

If  we  also  neglect  the  second  term  of  the  formula  just  given  —  as  it  evi 
dently,  for  the  present  purpose,  does  not  come  into  consideration  —  and, 
besides,  allow  ourselves  to  put 

z'  =  90°  —  y,  and  therefore  d  =  d, 

whereby  in  the  worst  case  only  so  much  will  be  lost  as  depends  upon 
the  polar  distance  of  the  Pole  Star,  we  receive 


=  D 

" 


-f  1)  .  sec  <p 
*  +  c  ..(tg  <?  +  sec  d  —  tg  8} 


which  expression  seems  quite  well  suited  to  serve  as  a  starting  point  for 
such  considerations.  We  will,  however,  here  content  ourselves  with  no 
ticing  only  that  which  has  reference  to  the  choice  of  the  star  which  we 
thus  far  have  spoken  of  under  the  name  of  the  Time  Star,  without  thereby 
having  intended  to  determine  anything  further  as  to  its  place  in  the 
heavens.  The  first  thought  is  to  direct  the  choice  by  preference  to  equa 
torial  stars  in  consideration  of  the,  in  itself,  correct  geometrical  principle, 
that  two  points  on  the  sphere  determine  the  corresponding  great  circle 
the  more  accurately  the  nearer  their  mutual  distance  approaches  a  right- 
angle  ;  to  which  the  further  reason  is  added,  that  for  the  equatorial 
stars  not  only  the  observed  transit  S,  but  also  the  assumed  known  right 
ascension  «,  are  liable  to  the  smallest  absolute  errors. 

On  closer  consideration,  however,  one  perceives  that  although  the 
latter  reason  here  may  certainly  have  some  weight,  on  the  other  hand 
the  geometrical  consideration  is  not  here  in  place  ;  that  is  to  say,  the 
two  points  that  determine  the  meridian,  i.  e.,  the  pole  and  zenith,  have, 
at  every  place,  a  given  distance  for  each  other  which  we  cannot  alter, 
and  upon  which,  of  course,  directly  depends  the  accuracy  with  which  the 
plane  of  the  meridian  can  in  any  way  be  determined.  But  the  determi 
nation  of  absolute  time  consists,  principally,  in  the  perception  of  that 
part  of  the  heavens  with  which  the  zenith  of  the  place  of  observation 
coincides  at  a  given  instant,  and  will,  therefore,  be  most  directly  attained 
by  the  observation  of  the  transits  of  zenith  stars.  This  is  already  shown 


THE    VERTICAL    OF    THE    POLE    STAR.  19 

in  that  for  zenith  stars  the  influence  of  the  azimuth  of  the  instrument 
disappears.  Our  formula  makes  the  relations  thus  brought  into  con 
sideration  still  plainer,  and,  if  the  necessary  data  were  known  with  suf 
ficient  accuracy,  would  even  allow  a  numerical  estimation  of  the  proba 
ble  errors  corresponding  to  the  different  cases.  But  also,  independent 
of  this,  we  see  that  on  account  of  the  factor  tg^> —  tgd,  the  influence  of 
an  error  in  the  angle  r  and  in  the  adopted  value  of  the  quantity  /*  dis 
appears  in  the  zenith,  whereby  it  deserves  to  be  particularly  mentioned 
that  the  latter  will  be  of  special  importance  in  the  application  of  a  mova 
ble  thread.  As  to  error  in  the  inclination  and  in  the  error  of  collima- 
tion,  the  former  influence  is  constant  for  all  stars;  the  latter  is  greater 
the  farther  the  star  is  removed  from  the  pole.  It  is,  therefore,  as  above 
already  remarked,  only  the  reliability  in  the  determination  of  D  which 
diminishes  with  increasing  inclination.  But  the  principal  law  of  this 
diminution  is  known  to  be  very  dependent  upon  the  peculiarity  of  the 
observer,  and  the  diminution,  in  general,  first  becomes  noticeable  at 
great  declinations,  so  that  finally  results  a  very  decided  advantage  for 
the  zenith  stars.  This  advantage  will  now  be  increased  by  the  circum 
stance  that  for  the  observation  in  the  zenith,  in  both  positions  of  the 
instrument,  the  same  portions  of  the  pivots  are  used ;  their  irregulari 
ties,  therefore,  are  completely  eliminated.  On  the  other  hand,  it  is  to 
be  allowed  that  the  greater  perfectness  of  the  instrument  in  and  of  itself, 
and  more  especially  the  shortening  of  the  duration  of  the  observation, 
essentially  diminishes  all  these  different  sources  of  error,  and  therefore 
gives  a  relatively  greater  weight  to  the  error  of  the  quantity  D,  which 
is  not  affected  by  them. 

On  taking  a  general  view,  therefore,  the  opinion  appears  authorized 
that  zenith  stars  certainly  have  an  advantage,  but  that,  in  our  method, 
this  is  less  important  than  in  the  establishment  in  the  meridian;  for  in 
this  latter  the  observation  of  zenith  stars  offers  the  only  means  of  free 
ing  ourselves  from  the  unwarranted  assumption  of  the  invariability  in 
azimuth,  while  in  respect  to  the  inclination,  the  possibly-existing  changes 
at  any  moment  can  be  recognized  by  means  of  the  level.  Such  a  greater 
freedom  in  the  choice  of  the  stars  to  be  observed  constitutes,  if  I  am  not 
mistaken,  a  further  not  unimportant  advantage  of  our  method. 

13.  It  now,  finally,  remains  to  mention,  with  a  few  words,  the  reduc 
tion  that  is  necessary  for  the  Time  Star,  if  this  is  observed  on  more  than 
one  thread.     Because  of  the  exhaustive  treatment  that  this  subject  has 
received  in  different  places,  especially  by  Hansen,  it  suffices  here  to 
merely  give  the  formulae  for  computation.     The  time,  f,  that  a  star  whose 
declination  is  d  needs  in  order  to  pass  from  any  great  circle  of  the  sphere 
to  the  parallel  circle,  distant /therefrom?  will  be  expressed  with  an  accu 
racy  entirely  sufficient  for  our  purposes  by  the  formula  : 

t  =f  V  sec  (d  +  n) .  sec  (d  —  n) , 

where  n  denotes,  as  previously,  the  distance  at  which  the  great  circle 
passes  by  the  pole.  A  knowledge  of  n,  sufficient  for  this  reduction,  is 
always  at  hand;  if  not  otherwise,  then  it  is  offered  by  our  computa 
tion  itself,  since  for  the  redaction  of  the  angle  r,  the  simple  passage  of 
the  Time  Star  through  the  middle  thread,  or  if  this,  perhaps,  is  not  ob 
served,  a  preliminary  reduction  of  the  side  thread,  with  the  factor  seed, 
will  suffice,  without  the  least  prejudice  to  the  accuracy  of  the  computa 
tion  5  and,  in  general,  the  exact  reduction  has  only  an  importance  if  the 
observed  threads  do  not  lie  symmetrical  with  respect  to  the  middle 
thread. 

14.  It  seems  proper,  in  conclusion,  to  demonstrate,  by  some  numerical 


20          THE  PORTABLE  TRANSIT  INSTRUMENT  IN 

example,  the  accuracy  and  convenience  of  the  above-given  formula  for 
computation.  As  a  first  example,  I  choose  that  given  by  Hansen  at  the 
conclusion  of  the  memoir  in  No.  199  of  the  AstronomischeNachrichteu. 
The  data,  according  to  the  notation  adopted  by  us,  are  as  follows : 

h.      m.         s.  °        '•          " 

a  =   9     59     18.86  d  =  12     47    33.6 

«'  =  18    27     22.5  (S'  =  86    35     19.9 

The  Time  Star  was  observed  on  three  threads,  the  Pole  Star  only  on 
the  third.  I  give  here  the  individual  transits,  together  with  the  corre 
sponding  thread  intervals : 

Time  Star.  Thread  interval.                       Pole  Star. 

h.      m.         s.  s. 

10    51     47.7  +39.50 

52  28.2  h.     m.       s. 

53  7.5  —38.30  11      5      51  =  S. 
Finally, 

<p  ==  500  56>  0" :     &  ==  _  3//.4  .     c  =  __  48.70  =  _  70//t6> 

We  content  ourselves  at  first  in  reference  to  S  with  the  observation 
of  the  middle  thread,  without  considering  the  two  others,  since  we  will 
not  yet  reduce  them  exactly  $  thus  we  have 

7i.         m.  8. 

S— «=D  =     0     53      9.34 
S'  —  a'==D'  =   16    38    28.5 


—  D=   15     45    19.16 

19'    47". 


We  will  now  at  first  compute  according  to  the  exact  formulae  of  article 
9,  in  order  to  test  thereby  the  .succeeding  computation  according  to  the 
approximate  formulae.  I  use  six-figure  logarithms  in  this  in  order  that 
nowhere,  by  reason  of  the  length  of  the  computation,  can  the  hundredths 
of  a  second  of  time  be  rendered  doubtful  through  accumulation  of  errors 
in  the  last  place  ;  but  I  give  the  computation,  not  completely,  but  only 
its  principal  step«3. 
A  The  formulae  I  give  : 

Q  I  II  O  /  " 

£  =  —2    53    25.6         d=+IQ    53     10.9 
thence  from  II  follows  :    rt  =  —       10    43.3 


therefore,  £  +  r,  =  —  3      4      8.9 

And  with  this,  according  to  III : 

x  =  —  0    39     39.87      n=  —   2    59    50.0 
finally,  from  IV :  m  =  —  3    41     59.69 

therefore,  x  —  m  =  +  3      2    19.82 

With  the  ?t,  thus  found,  follows 

log  V  sec  (d  +  n) .  sec  (d  —  n)  =  0.01154, 

and  with  this  the  reduction  for  the  time  stars  to  the  middle  thread  be 
comes  4-  40S.56  and  —  39S.33.  The  three  transits  through  the  middle 
thread  are,  therefore, 


THE    VERTICAL    OF    THE    POLE    STAR.  21 


therefore,  more  accurately, 
and 

x  — 

Rnr  WA  nQ.Ara 

h. 

10 

m. 
52 

8. 

28.56 

28.2 
28.17 

8=10 
D=  0 

m 

52 
53 

19 

28.21 
9.35 

Q  QO 

therefore,  finally,  u=    — 41      0.03 

For  the  computation,  according  to  the  formulae  of  article  11,  now  to 
be  given,  I  borrow  from  the  foregoing  the  value  of 

o  / 

d=W    53.2 

Then  is  00°  —  d  =  z'  +  z=  79       6.8 

further,  <p  —  d  =  z  =  38      8.4 


whence,  z'=  40    58.4 

which  I  assume  to  have  been  read  from  the  finding  circle  or  taken  from 
an  ephemeris  of  the  Pole  Star.  As  above  remarked,  these  quantities 
are  only  necessary  in  the  computation  of  the  coefficients  C  and  F.  In 
respect  to  the  former  I  remark,  that,  in  general,  not  its  logarithm  but 
the  number  itself  will  be  first  used,  at  least  in  case  the  error  of  collima- 
tion  is  not  assumed  as  known,  but  is  first  to  be  deduced  from  the  ob 
servations  themselves.  In  such  cases  the  above-given  expression  for  C 
is  quite  convenient,  especially  if  we  observe  that  one  portion  of  it  recurs 
again  in  F.  In  other  cases,  as  therefore  in  the  present,  it  is  more  con 
venient  to  change  it  into 

0  =  sec  (p  cos  (  z  "~     i  sec 


As  regards  the  term  B  ft,  we  sliould  not  hesitate  to  connect  its  compu 
tation  with  the  equally  necessary  conversion  of  the  level  divisions,  as 
directly  given,  into  the  corresponding  value  in  arc  5  that  is  to  say,  we 
seek  not 

1)  =  —  3".4,  but  at  once  B  b  =  ~  .  sec  <p  =  —  Os.360. 

lo 

From  the  given  values  of  z1  and  z  follows 

o  // 

?-t?=:<r=39     33.4 
Z-.=  A  =   1     25.0 


and,  with  these,  the  complete  computation  of  the  terms  C  c  and  F/  be 
come  the  following  : 

log  cos  A  =  9.99987  log  00  =  0.3133  log  F0  =  9.99909 

log  sec  o  =  0.11295  log  c   =  0.6721ra  log  /  =  1.58320& 


log  sec  <p  =  0.20050  log  C0  c  =  0.9854/&          log  F0/  =  1.58229& 

log  sin  z  =  9.79070      q  .  log  sec  m  =  0.0015      Jc  log  sec  m  =  0.00227 

log  sec  d  =  0.00789  log  C  c  =  0.9869^  log  F/  =  1.58456/1 


22  THE    PORTABLE    TRANSIT    INSTRUMENT    IN 

The  corrections  dependent  upon  sec  m  have  been  later  introduced,  after 
obtaining  the  values  of  <j.  and  in.  Their  omission  would  have  introduced 
an  error  of  only  Os.033  in  C  c  but  of  Os.200  in  F/. 

I  have  thus  previously  executed  the  computation  of  the  terms  C  c  and 
F/,  because,  as  above  remarked,  in  the  actual  application  of  our  method, 
I  suppose  the  factors  C  and  F  to  be,  once  for  all,  computed  and  brought 
into  a  table.  I  will  hereafter  give  a  portion  of  such  a  table.  The  fol 
lowing  is  the  computation  that  is  now  to  be  independently  executed  for 
each  separate  observation : 

o   .      /  // 

log  tg?  =  0.090598  #o  =  —         38    25.90 

log  tg  3  =  9.356140  m0  =  —   3    28    38.63 


log  cotg<5'  =  8.775291  x0  —  m0  =  +    2    50     12.7 

log  denominator   =   —  3247 


s. 


.  . 

log  sin  r  =  9.920250ft                     ?>  =  +  53      9.35 

logA  =  8.131431                     B6  =  —  0.360 

log  COST  =  9.74383^                     Cc  =  —  9.703 

Argum  =2.12474  <r                    ~Pf=—  38.420 

log  fi  =  0.734458  +  52    20.83 

=  11    20.85 


lo 

log  cos  #0=       —27  u  =  —         40     59.98 

log  sin  m0  =  8.782865^ 

that  is  to  say,  differing  by  Os.05  from  the  result  of  the  exact  computa 

tion.     Now,  it  is  not  difficult  to  see  where  the  cause  of  this  discordance 

is  to  be  sought.     By  reason  of  the  magnitude  of  the  term  F/,  it  is  cer 

tainly  not  surprising  that  the  terms  of  the  higher  order  neglected  therein 

have  not  been  entirely  evanescent.    But  instead  of  developing  these,  it 

x     is  certainly  much  more  convenient,  in  such  a  case,  to  ^ex^cutejthe  compu- 

^VXtation  rigorously  as  regards  J^  and  tbe^thereby  m&£m&=3sbo?  proves 

^J    itself  to  btTso  slight,  especially  in  consideration  that  the  F/  now  en 

tirely  falls  out,  that  it  seems  advisable  always  to  choose  this  solution  if 

the  factor  F  is  not  already  elsewhere  given.    The  system  of  equations 

will  be  as  follows  : 

,    ^  _      sec  d  cotg  d1  sin  r 

~  1  —  tg  o  COtg  d1  COS  r 

sin/ 


sin  mi  =  cos  d  tg  (£  +  rt]  tg  <p  cos  #1 

ii  =f  5L_^  —  (B  +  B&  +  Cc) 


These  being  applied  to  our  example,  the  following  computation  results, 
which  I  here  give  without  any  omission  : 

o        /  // 

log  tg  6  =  9.356140  log  /=  2.75929^  f  =  —  2    53    25.5? 

log  cotg  <i'=  8.775291    log  sin  (*'  +  *)  =  9.99211  rj  =  —         9     45.03 


log  sec  d  =  0.010916        log  n  =  2.76718w      ^^  =  —  3   3  10.6 
log  tg  6  cotg  6'  =  8.13143       log  sin  d  =  9.345224 


THE    VERTICAL    OF    THE    POLE    STAR.  23 

o         /  // 

logcosr=9.74383n     log  tg  (sc+  ?)  =  8.727007/1  xi=  40    35.77 

Arguin.  =2.12474(7  log  cos  6  =  9.989084  mi  =  —3    40    24.58 

log  seed  cotg  <J'=  8.786207  log  tg  0  =  0. 090598  JT]  —  »HI  =  +  2    59     48.81 

m.         s. 
log  shir  =  9.920250w  logcosjci=        —30  '  ~  m\  =  -f        11     59.254 

J-O 

log  denominator  =    —3247  log  tg  xl  =  8.072231  n  D-f-B6+Cc=+        52    59.287 

log  tgf  =  8.703210»          log  sin  •)»!  =  8.806659»  11=  41       0.03 

15.  The  previous  example  is  treated  with  abundant  fullness,  in  order 
to  leave  no  doubts  as  to  the  way  to  be  pursued  in  all  cases  that  can 
occur.  It.  is,  however,  in  fact  more  unfavorably  conditioned  than  is 
necessary  to  assume  in  actual  practice.  First,  «  Ursa?  Minoris  should 
be  always  observed ;  and,  secondly,  a  collimation  error  of  nearly  five 
seconds  of  time  is  an  aggravation  that  can  be  avoided  with  s  ome  atten 
tion.  The  example  is,  also,  evidently  only  a  fictitious  case.  I  will  now,  by 
the  computation  of  an  observation  actually  made,  and  noways  specially 
favorable,  show  the  process  that  we  are  accustomed  to  follow  in  the 
treatment  of  this  problem.  To  this  end  we  have  computed,  once  for  all, 
the  coefficients  0  and  F,  for  our  latitude  59°  46'  20",  for  a  number  of  the 
more  frequently  observed  stars,  under  the  assumption  that  the  Time  Star 
is  observed  some  six  minutes  after  the  Pole  Star,  which  corresponds  very 
nearly  to  the  interval  that  occurs  in  the  actual  observations.  In  the 
example  here  to  be  considered  the  following  come  into  use : 

C  logC  logF 

13  Draconis     -     -     -     -     2.060  0.3139  9.6151 

r  Draconis     -     -     -     -    2.069  0.3158  9.6576 

a  Lyr« 2.201  0.3427  9.9613 

C  Aquilse  -  2.507  0.3997  0.1688 

In  reference  to  the  computation  itself,  I  make  now,  further,  the  follow 
ing  remarks: 

For  3  =  0  the  coefficient  v  attains  an  infinite  value  ;  since,  however,  at 
the  same  time  A  becomes  —  0,  our  equations  in  this  case  are 

tg  x0  =  0  sin  m0  ==  tg  <p  cotg  $'  sin  -. 

But  this  indicates,  that  in  order  to  be  able,  in  all  cases,  to  conduct  the 
computation  according  to  quite  similar  ways,  whereby  it  is  known  that 
in  the  actual  application  no  slight  relief  is  attained,  we  must  subject  our 
formulae  still  to  a  small  change.  We  put  ^ 

A  .  p.  =  tg  <?  COtg  V  =  v 

and  Sinr        =, 

1  —  A  COS  r 

then  will  tg  XQ  =  A  .  p 

and  sin  m0  =  v  .  p  .  cos  x0 ; 

and  the  further  advantage  is  here  connected  that  the  factor  v  can  be 
considered  as  constant  for  all  observations  of  the  same  evening. 

Since  we  always  observe  the  Pole  Star,  the  entire  angle  to  be  computed 
does  not  reach  3°  even  for  a  latitude  of  60°.  For  such  angles  as  is  well 
known,  the  excellent  five-figure  logarithmic  tables  of  Westphal,  by  the 
use  of  the  small  auxiliary  tables  headed  "Corr.,"  in  connection  with  the 
logarithms  of  the  numbers,  make  the  use  of  the  trigonometric  tables 
quite  unnecessary :  and  it  may  be  hereremarked  that  we  have,  with 

A- 


24          THE  PORTABLE  TRANSIT  INSTRUMENT  IN 

great  regret,  missed  this  addition  in  the  otherwise  perfect  tables  of 
Bremiker.  On  this  account,  principally,  we  confine  ourselves  to  five 
decimals  in  this  computation,  although  thereby,  under  some  circum 
stances,  the  hundredths  of  a  second  of  time  certainly  become  unsafe; 
but  this  can  be  considered  as  no  noticeable  diminution  of  accuracy  in 
the  observation  of  the  transit  of  a  single  star. 

As  to  the  following  observations,  they  have  been  made  by  a  talented 
and  industrious  young  officer,  Mr.  Koverski,  who  at  present,  as  a  pupil 
of  the  military  academy,  takes  part  in  the  two-year  practical  course  at 
Poulkova;  and,  indeed,  at  that  time  he  had  had  an  experience  of  only 
the  first  few  weeks  in  such  observations.  The  instrument  is  an  Ertel 
portable  transit  instrument  of  larger  dimensions,  but  with  the  above- 
mentioned  erroneous  position  of  the  clamp  screw.  The  reticule  consists 
of  nine  threads,  and  we  are  accustomed  to  indicate  these  with  the  num 
bers  I  to  IX  in  fixed  order,  namely,  according  to  their  increasing  dis 
tances  from  the  ocular  end  of  the  axis.  The  otherwise  customary  nota 
tion,  according  to  the  order  in  which  the  observed  stars  pass  through 
them,  becomes  ambiguous  when  the  Pole  Star  stands  very  near  its  elonga 
tion.  Small  marks  upon  the  threads  themselves  make  a  confusion  im 
possible  in  our  method  of  numbering  5  and  this  is  important,  since  in  the 
observation  of  the  Pole  Star  upon  only  one  thread  an  error,  in  this  re 
spect,  would  be  very  dangerous.  In  the  use  of  a  moveable  thread,  the 
numbers  upon  the  micrometer  are  also  quite  as  safe  a  means  of  indicating 
the  fixed  threads.  The  distances  from  the  middle  thread,  that  hasfcome 
into  use,  are  the  following : 

VI  =  5S.731,  VII  =  17S.701. 

These  refer  to  the  first  of  our  examples,  in  which  the  observations  in 
both  positions  are  made  in  the  same  azimuth  ;  while  in  the  second,  the 
Pole  Star  was  too  near  the  elongation,  and  therefore  by  changing  the 
azimuth  of  the  instrument,  it  was  each  time  observed  upon  the  middle 
thread.  The  following  table  contains  the  entire  computation,  exactly  in 
the  form  in  which  we  always  conduct  it.  It  needs  certainly  scarcely 
any  further  explanation,  except  that  the  factor  sin  V  is  omitted  on  the 
lines  distinguished  by  the  angular  brackets.  The  star  positions  are 
those  of  the  British  Xautical  Almanac.  The  constants  for  the  evening 
are: 

a'  =  lh  9m  43s 
log  cot  8'  =  8.39474 
log  v  =  8.62933 


THE   VERTICAL    OF    THE    POLE    STAR. 


25 


Date. 

1863,  July  29. 

1863,  July  29. 

Position. 

East, 

West. 

West. 

East. 

Time  star. 

j3  Draconis. 

y  Draconis. 

a  Lyrae. 

£  Aquihe. 

h.  m.       8. 

7*.  m.      8. 

/(.  m.       s. 

h.  m.       8. 

g/   , 

17  25    4    VI 

17  55    2   VII 

18  34  44      M 

19    4  20    M 

S 

17  32  44.55 

17  59    9.69 

18  40  51.  39 

19  10  45.44 

a 

17  27  23.  05 

17  53  28.  54 

18  32  21.  35 

18  59  10.47 

D 

16  15  21 

16  45  19 

17  25     1 

17  54  37 

o 

0    5  21.50 

0    5  41.  15 

0     8  30.  04 

0  11  34.97 

—    0.50 

—    0.18 

—    0.15 

—  0.27 

F/ 

—    2.36 

4     8.05 

0.00 

0.00 

h.     m.     8. 

h.   m.     s. 

n.  m.  s. 

h.   m.     s. 

D'  —  D 

16    9  59.5 

16  39  37.  8 

17  16  3l'.  0 

17  43    2.0 

o         / 

o         / 

O              ' 

o         / 

T 

242  29.9 

249  54.5 

259    7.8 

260  45.5 

(5 

52  24.5 

51.  30.  6 

38  39.  8 

13  40.0 

log  tg  6 
log  sin  r 

0.  11358 
9.  94792H 

0.  09955 
9.  97274w 

9.  90314 
9.  99214* 

9.  38589 
9.  99881« 

log  denominator 

—    642 

—    463 

—     162 

—     19 

Argument 

1.  82725(7 

1.96975(7 

2.  42662a 

3.  3504(7 

log  cos  r 

9.  66443  n 

9.  53596» 

9.  27550/t 

8.  8690  M 

& 

log  A 

8.  50832 

8.  49429 

8.  29788 

7.  78063 

[log  p] 

5.  25593/< 

5.  28254-w 

5.  30495w 

5.  31305/* 

log  cos  x0 
[logtgzo] 
[log  sin  ™0] 

—     17 
3.  76425w 
3.  88509/j 

—     18 
3.  77683/j 
3.  91169» 

—      8 
3.  60283« 
3.  93420w 

—      1 
3.09368H  • 
3.  94237  » 

0       /            // 

o     /         // 

O        /            /' 

0       /           " 

X 

—      1  36  49.5 

—       1  39  40.  1 

—      1     6  46.6 

—  0  20  40.7 

/HO 

—      2    7  57.0 

—      2  16    2.0 

—      2  23  16.  6 

—  2  26    0.  0 

X0  —  /»o 

-|-       0  31     7.  5 

4      0  36  21.  9 

4       1  16  30.  0 

42    5  19.3 

m.     s. 

m.      s. 

7/1.        8. 

W.       8. 

i_  /^  Wo) 

4          2    4.  50 

4-          2  25.  46 

{-          5    6.00 

4      8  21.29 

i>4B&4°F/ 

4          5  18.64 

-j-          5  49.02 

4          8  29.  89 

4     11  34.70 

w  +  Cc 

—          3  14.  14 

3  23.  56 

3  23.89 

—      3  13.41 

Clock  rate 

0.04 

4                0.04 

0.04 

4            0.04 

The  chronometer  gained  48.0  in  twenty-four  hours'  sidereal  time,  and 
corresponding  to  this  rate  are  the  reductions  to  the  mean  mom  nit  of 
each  pair  of  determinations,  as  given  in  the  last  line  of  the  individual 
clock  corrections.  Such  a  reduction  is  necessary  to  the  exact  deduction 
of  the  collimation  error.  If  we  indicate  this  by  -f  c  for  position  west, 
and  therefore  —  c  for  the  position  east,  we  have  now,  by  means  of  the 
coefficients  0,  the  following  determinations  : 


m.      s. 

w_  2.060.  c  =  —  3  14.18 
u  +  2.069  .  c  =  —  3  23.52 
therefore,         +  4.129.  c=        —9.34 
and  thence.  c=        —  2.262 

u  =  —  3  18.84 
for  17h  46 


. 

M  +  2.201.  c=  -3  23.93 
u  —  2.507  .  c  =  —  3  13.37 
4.4.708.6'=:—  10.56 
c  =  —   2.243 
u  =  —  3  18.99 
for  18h  56m 


The  agreement  of  the  two  values  of  c  is  thoroughly  satisfactory,  and 
the  change  in  value  of  u  corresponds,  within  Os.04,  to  the  above-given 
rate  of  the  chronometer. 


26 


THE  PORTABLE  TRANSIT  INSTRUMENT  IN 


TABLE  I. 
FINDING  EPHEMERIS  OF  POLARIS.— J  =  88°  37'. 


0=+20°. 

0=+30°. 

0=^40°.     0=+50o. 

0=4-60°. 

<fr=  +  70°. 

Hour 

Hour 

angle. 

angle. 

Z. 

A. 

Z. 

A. 

Z. 

A. 

Z. 

A. 

Z. 

A. 

Z. 

A. 

k. 

o  / 

0   / 

o  / 

o  / 

o  / 

0   / 

o  / 

o  / 

o  / 

o  / 

o  / 

o  / 

h. 

0 

68  37 

0  0 

58  36 

0  0 

48  37 

0  0 

38  36 

0  0 

28  37' 

0  0 

18  37 

0  0 

24 

1 

40 

23 

39 

24 

40 

28 

39 

34 

40 

0  45 

39 

1  7 

23 

2 

48 

44 

48 

48 

48 

55 

47 

1  6 

49 

1  26 

48 

2  9 

22 

3 

69  1 

1  3 

59  2 

1  8 

49  1 

1  18 

39  1 

33 

29  3 

2  1 

19  2 

3  0 

21 

4 

19 

17 

18 

24 

18 

35 

18 

53 

20 

27 

20 

3  37 

20 

5 

39 

26 

39 

33 

39 

45 

39 

2  6 

40 

42 

40 

3  58 

19 

6 

70  0 

1  28 

60  0 

36 

50  0 

48 

40  0 

9 

30  2 

46 

20  1 

4  3 

18 

7 

21 

25 

21 

32 

21 

44 

21 

2  4 

20 

39 

24 

3  50 

17 

8 

41 

16 

42 

22 

42 

34 

42 

1  50 

39 

21 

42 

3  23 

16 

9 

59 

1  2 

59 

1  7 

59 

1  16 

59 

30 

30  56 

1  54 

20  59 

2  44 

15 

10 

71  12 

0  44 

61  12 

0  47 

51  12 

0  53 

41  19 

1  3 

31  10 

1  20 

21  11 

1  55 

14 

11 

20 

23 

21 

24 

21 

0  28 

21 

0  32 

20 

0  41 

20 

0  59 

13 

12 

23 

0  0 

24 

0  0 

23 

0  0 

24 

0  0 

23 

0  0 

22 

0  0 

12 

A  is  to  be  added  to  180°  to  obtain  the  azimuth  when  the  hour  angle  is  >12h. 

A  is  to  be  subtracted  from  180°  to  obtain  the  azimuth  when  the  hour  angle  is  <l2b. 


TABLE  II. 
COEFFICIENT  OF  AZIMUTHAL  CHANGE  FOR  A  UNIT  OF  HOUR  ANGLE. 


S     0=+20. 

30. 

40. 

50. 

60.         70. 

i 

o 

i 

o 

—  50 

+     0.68 

-  - 

.  . 

—   50 

40 

0.81 

+     0.82 

40 

30 

1.13 

1.00 

+     6.92 

30 

20 

1.46 

1.23 

1.08 

+    i.  06 

20 

—  10 

1.97 

1.53 

1.28 

1.14 

+     1.  05 

10 

0 

2.92 

2.00 

1.56 

1.30 

1.15  '+     1.06 

—   0 

+   10 

+     5.  67 

2.88 

1.97 

1.53 

1.  28        1.  14 

+  10 

20 

oo 

4-     5.41 

2.75 

1.88 

1.  46  !       1.  23 

20 

30 

4.99 

CO 

+     4.99 

2.53 

1.  73  !       1.  35 

30 

40 

2.14 

4.41 

OO 

+     4.41 

2.  24        1.  53 

40 

50 

1.28 

1.88 

3.70 

OO 

+     3.70        1.88 

50 

60 

0.78 

1.00 

1.46 

—     2.88 

CO   '  +     2.88 

60 

70 

—     0.45 

—     0.53 

0.68 

1.00 

1.  97  ;       CO 

70 

110  ! 

+     6.35 

+     6.36 

+     6.40 

+     0.45  |+     6.53 

110 

120  [ 

+     0.51 

0.53 

0.  58  I      0.  65 

120 

130 

+     0.65 

0.  68        0.  74 

130 

140 

+     0.  78  !      0.82 

140 

+  150        .  . 

.  .   +     0.88 

+  150 

THE    VERTICAL    OF    THE    POLE    STAR. 


27 


TABLE  III. 


Angle. 

S. 

T. 

Angle. 

S. 

T. 

/; 

, 

// 

0   1 

0 

0 

0 

0 

3600 

1  0 

22 

44 

60 

I 

0 

0 

3660 

1  1 

23 

45 

120 

2 

0 

0 

3720 

1  2 

24 

47 

180 

3 

0 

0         3780         1  3 

24 

48 

240 

4 

0 

0         3840 

1  4 

25 

50 

300 

5 

o 

.  0         3900 

1  5 

26 

52 

360 

6 

0 

0         3960 

1  6 

27 

53 

420 

7 

.0 

0 

4020 

1  7 

28 

55 

480 

8 

1 

1 

4080 

1  8 

28 

57 

540 

9 

1 

1 

4140 

1  9 

29 

58 

600 

10 

1 

1 

4200 

1  10 

30 

60 

660 

11 

1 

1 

4260 

1  11 

31 

62 

720 

12 

1 

2 

4320 

1  12 

32 

63 

780 

13 

1 

2 

4380 

1  13 

33 

65 

840 

14 

1 

2         4440 

1  14 

34 

67 

900 

15 

2 

3         4500 

1  15 

35 

69 

960 

16 

2 

3         4560        1  16 

36 

71 

1020 

17 

2 

3 

4620        1  17 

36 

73 

1080 

18 

2 

4 

4680        1  18 

37 

74 

1140 

19 

2 

4 

4740        1  19 

38 

76 

1200 

20 

3 

5         4800        1  20 

39 

78 

1260 

21 

3 

5 

4860        1  21 

40 

80 

1320 

22 

3 

6         4920        1  22 

41 

82 

1380 

23 

3 

6         4980        1  23 

42 

84 

1440 

24 

4 

7         5040        1  24 

43 

86 

1500 

25 

4 

8         5100 

1  25 

44 

88 

1560 

26 

4 

8 

5160 

1  26 

45 

90 

1620 

27 

5 

9 

5220 

1  27 

46 

93 

1680 

28 

5 

9 

5280 

1  28 

48 

95 

1740 

29 

5 

10 

5340 

1  29 

49 

97 

1800  ' 

30 

6 

11         5400 

1  30 

50 

99 

1860 

31 

6 

12 

5460 

1  31 

51 

101 

1920 

32 

6 

12 

5520 

1  32 

52 

104 

1980 

33 

7 

13 

5580 

1  33 

53 

106 

2040 

34 

7 

14 

5640 

1  34 

54 

108 

2100 

35 

8 

15 

5700 

1  35 

55 

110 

2160 

36 

8 

16 

5760 

1  36 

57 

113 

2220 

37 

9 

17 

5820        1  37 

58 

115 

2280 

38 

9 

18 

5880        1  38 

59 

118 

2340 

39 

9 

18 

5940        1  39 

60 

120 

2400 

40 

10 

19         6000 

1  40 

61 

122 

2460 

41 

10 

20         6060 

1  41 

63 

125 

2520 

42 

11 

21 

6120 

1  42 

64 

'  127 

2580 

43 

11 

28 

6180 

1  43 

65 

130 

2640 

44 

12 

24 

6240        1  44 

66 

132 

2700 

45 

13 

25 

6300 

1  45 

68 

135 

2760 

46 

13 

26 

6360 

1  46 

69 

138 

2820 

47 

14 

27 

6420 

1  47 

70 

140 

2880 

48 

14 

28 

6480 

1  48 

72 

143 

2940 

49 

15 

29 

6540 

1  49 

73 

145 

3000 

50 

15 

30 

6600        1  50 

74 

148 

3060 

51 

16 

32         6660        1  51 

76 

151 

3120 

52 

17 

33 

6720 

1  52 

77 

154 

3180 

53 

17 

34 

6780 

1  53 

78 

156 

3240 

54 

18        36 

6840 

1  54 

80 

159 

3300 

55 

19 

37 

6900 

1  55 

81 

162 

3360 

56 

19 

38 

6960 

1  56 

83 

165 

3420 

57 

20 

40 

7020 

1  57 

84 

168 

3480 

58 

21 

41 

7080 

1  58 

85 

170 

3jd40 

59 

21 

43 

7140 

1  59 

87 

173 

3600 

60 

22        44 

7200 

1  60 

88 

176 

THE  PORTABLE  TRANSIT  INSTRUMENT,  ETC. 


TABLE  III.— Continued. 


Angle. 

S. 

T. 

Angle. 

S. 

T. 

„ 

o  / 

M 

o  / 

7200 

2  0 

88 

176 

10800 

3  0 

199 

397 

7260 

2  1 

90 

179 

10860 

3  1 

201 

401 

7320 

2  2 

91 

182 

10920 

3  2 

203 

406 

7380 

2  3 

93 

185 

10980 

3  3 

205 

411 

7440 

2  4 

94 

188 

11040 

3  4 

207 

415 

7500 

2  5 

96 

191 

moo 

3  5 

419 

7560 

2  6 

9? 

194 

11160 

3  6 

212  ' 

424 

7620  . 

2  7 

99 

198 

11220 

3  7 

214 

428 

7680 

2  8 

100 

201 

11280 

3  8 

217 

433 

7740 

2  9 

102 

204 

11340 

3  9 

219 

438 

7800 

2  10 

104 

207 

11400 

3  10 

221 

442 

7860 

2  11 

105 

210 

11460 

3  11 

223 

447 

7920 

2  12 

107 

213 

11520 

3  12 

227 

452 

7980 

2  13 

108 

217 

11580 

3  13 

229 

456 

8040 

2  14 

110 

220 

11640 

3  14 

231 

461 

8100 

2  15 

112 

223 

11700 

3  15 

233 

466 

8160 

2  16 

113 

227 

11760 

3  16 

235 

471 

8220 

2  17 

115 

230 

11820 

3  17 

237 

476 

8280 

2  18 

117 

233 

11880 

3  18 

240 

481 

8340 

2  19 

118 

237 

11940 

3  19 

242 

486 

8400 

2  20 

120 

240 

12000 

3  20 

245 

490 

8460 

2  21 

122 

243 

12060 

3  21 

247 

495 

8520 

2  22 

124 

247 

12120 

3  22 

250 

500 

8580 

2  23 

125 

250 

12180 

3  23 

252 

505 

8640 

2  24 

127 

254 

12240 

3  24 

255 

510 

8700 

2  25 

129 

258 

12300 

3  25 

257 

515 

8760 

2  26 

131 

261 

12360 

3  26       260 

521 

8820 

2  27 

132 

265 

12420 

3  27 

263 

525 

8880 

2  28 

134 

268 

12480 

3  28 

266 

530 

8940 

2  29 

136 

272 

12540 

3  29 

268 

535 

9000 

2  30 

138 

276 

12600 

3  30 

271 

540 

9060 

2  31 

140 

279 

,  12660 

3  31 

273 

545 

9120 

2  32 

142 

283 

%/  1)720 

3  32 

275 

551 

9180 

2  33 

144 

287 

/  12780 

3  33 

278 

556 

9240 

2  34 

145 

291 

y   12840 

2  34 

281 

561 

9300 

2  35 

147 

294 

12900 

3  35 

284 

566 

9360 

2  36 

149 

298 

12960 

3  36 

286 

572 

9420 

2  37 

151 

302 

13020 

3  37 

289 

577 

9480 

2  38 

153 

306 

13080 

3  38 

291 

582 

9540 

2  39 

155 

310 

13140 

3  39 

294 

588 

9600 

2  40 

157 

314 

13200 

3  40 

297 

593 

9660 

2  41 

159 

318 

13260 

3  41 

300 

598 

9720 

2  42 

161 

322 

13320 

3  42 

302 

604 

9780 

2  43 

163 

325 

13380 

3  43 

305 

610 

9840 

2  44 

165 

330 

13440 

3  44 

307 

615 

9900 

2  45 

167 

334 

13500 

3  45 

310 

621 

9960 

2  46 

169 

338 

13560 

3  46 

313 

626 

10020 

2  47 

171 

342 

13620 

3  47 

316 

632 

10080 

2  48 

173 

346 

13680 

3  48 

318 

637 

10140 

2  49 

175 

350 

13740 

3  49 

321 

643 

10200 

2  50 

177 

354 

13800 

3  50 

324 

649 

10260 

2  51 

179 

358 

13860 

3  51 

327 

654 

10320 

2  52 

181 

362 

13920 

3  52 

330 

660 

10380 

2  53 

183 

367 

13980 

3  53 

333 

666 

10440 

2  54 

186 

371 

14040 

3  54 

335 

671 

10500 

2  55 

188 

375 

14100 

3  55 

338 

•  677 

10560 

2  56 

190 

380 

14160 

3  56 

341 

683 

10620 

2  57 

192 

384 

14220 

3  57 

34*  tfl 

688 

10680 

.   2  58 

194 

388 

14280 

3  58 

347  ' 

695 

10740 

2  59 

196 

393 

14340 

3  59 

350  ,' 

701 

10800 

2  60 

199 

397 

14400 

3  60 

352 

707 

APPENDIX    AND    TABLES, 

BY  THE  TKANSLATOB. 


APPENDIX. 


The  memoir  here  presented  to  the  readers  notice  was  published  in 
1863  by  the  Imperial  Central  Observatory  at  Poulkova,  (near  St.  Peters 
burg,)  after  the  methods  herein  given  had  been  carefully  elaborated  by 
their  author  and  often  tested  in  the  course  of  his  experience  as  senior 
astronomer  at  that  institution  and  as  professor  of  geodesy  to  the  Impe 
rial  Military  Academy. 

Until  1864  the  practical  application  of  Dolleii's  method  was  seriously 
embarrassed,  inasmuch  as  no  instrument  specially  proper  to  this  class 
of  observations  had  been  constructed,  and  the  use  of  u  universal  instru 
ments"  and  Hansen's  formulae  could  only  be  said  to  have  given  promise 
of  what  would  be  achieved  with  a  suitable  instrument.  In  the  spring 
of  this  year,  however,  there  were  finished  by  Brauer,  the  then  mechani 
cian  of  the  observatory,  three  portable  transits,  two  of  which  were  de 
signed  for  the  immediate  use  of  the  observers  engaged  in  determining 
the  longitudes  of  points  on  the  Yaleutia-Orsk  parallel ;  the  third,  soon 
after  its  completion,  passed  into  the  hands  of  Professor  Schidloifski, 
director  of  the  observatory  at  Kieff,  who  took  it  with  him  on  returning 
home  in  August  from  the  quarter-century  inauguration  anniversary  at 
Poulkova. 

This  latter  instrument  it  was  that,  having  been  during  the  summer  at 
the  disposal  of  Mr.  Dollen,  would,  as  he  hoped,  enable  him  to  carry  out 
such  special  studies  as  would  decide  definitively  upon  the  advantages 
resulting  from  its  use,  the  instrument  having  been  made  by  Brauer, 
under  his  own  supervision,  and  with  direct  reference  to  the  convenient 
application  of  his  method  for  the  determination  of  time. 

But  the  series  of  observations  then  begun,  unfortunately,  could  not  be 
completed,  and  they  still  remain  unpublished,  and  probably  not  fully 
discussed — a  delay  necessitated  by  the  pressure  of  imperative  duties 
consequent  upon  Dollen's  appointment  ad  interim  to  the  directorship  of 
the  observatory,  and  an  illness  that  for  months  threatened  his  life,  and 
finally  compelled  him  to  a  year's  absence  from  Poulkova. 

In  the  summer  of  1864  Messrs.  Thiele  and  Zylinski,  engaged  on  the 
above-mentioned  arc  of  longitude,  being  in  England,  their  instruments 
were  brought  so  favorably  to  the  notice  of  the  astronomer  royal  that 
Professor  Airy  ordered  from  Brauer  a  similar  portable  transit  for  the 
Greenwich  Observatory.  This  was  finished  in  the  winter  of  1865-'66, 
and  the  accuracy  of  its  construction  was  sufficiently  tested  during  a  few 
clear  days  in  the  following  spring,  the  writer  having  himself  improved 
the  opportunity  of  examining  the  cyliudricity  of  its  pivots  by  means  of 
the  microscope  apparatus  supplied  with  it. 

Simultaneously  with  the  construction  of  the  latter  instrument  Brauer 
took  in  hand  two  others,  (Nos.  Y  and  VI,)  which  he  was  not  then  able 
to  complete,  owing  to  his  other  engagements  and  the  removal  of  his 
workshop  to  St.  Petersburg,  but  which  he  has  promised  shall  be  even 
superior  to  their  predecessors,  and  which,  if  not  now  finished,  certainly 
can  be  within  a  few  months. 


32  THE    PORTABLE    TRANSIT    INSTRUMENT    IN 

THE  BRATJER  PORTABLE   TRANSIT. 

The  Brauer  portable  extra-meridional  transit  instrument  consists,  as 
to  its  chief  features,  of  an  objective  of  2.5  to  3.0  inches  aperture  and  30 
to  36  inches  focal  length,  bearing  a  power  of  150.  The  ocular  being  at 
one  end  of  the  axis  of  revolution,  the  light  for  the  field  illumination 
enters  through  the  opposite  pivot ;  the  total  reflection  prism  within  the 
central  cube  admits  of  proper  adjustment.  To  the  stationary  reticule  of 
five  wires  is  added  a  pair  of  close  double  wires,  moved  by  a  fine  microme 
ter  screw.  The  hanging  level  need  never  be  removed  from  its  position 
on  the  pivots.  An  unusually  expeditious  reversing  apparatus  is  always 
in  place,  and  works  with  scarcely  any  possibility  of  detriment  to  the  sta 
bility  of  azimuth  or  level.  This  apparatus  is  so  contrived  as  not  to  inter 
fere  with  nadir  observations,  made  by  means  of  a  dish  of  mercury  placed 
on  the  pier  and  below  the  base  of  the  tripod.  The  three  foot  screws  rest 
on  corresponding  blocks,  one  of  which  is  itself  moveable  by  a  horizontal 
adjusting  screw,  allowing  the  instrument  to  be  accurately  placed  at  any 
azimuth  within  a  range  of  six  or  seven  degrees.  Microscopes  and  levels 
for  the  examination  of  the  pivots  are  also  provided. 

The  excellence  of  these  instruments  depends  so  much  upon  the  con 
venient  arrangement  and  conscientious  workmanship  of  all  the  parts, 
that  actual  use  and  the  critical  study  of  the  results  can  alone  (as  it  act 
ually  does)  suffice  to  persuade  one  of  their  superior  merits. 

While  the  Brauer  transit  is  unusually  convenient  for  use  in  any  verti 
cal  plane,  it  may,  of  course,  be  very  advantageously  mounted  in  the  me 
ridian  ;  it  is  in  this  way  that  the  International  Commission,  having  the 
Yalentia-Orsk  arc  in  charge,  have  decided  to  employ  it,  preferring,  it 
would  seem,  to  sacrifice  time  rather  than  risk  the  introduction  of  un 
known  errors  by  the  adoption  of  new  methods  in  a  work  of  such  magni 
tude  and  importance.  That  they,  in  1863,  may  have  been  justified  in 
this  discussion  will  be  admitted,  but  their  own,  as  well  as  Dollen's,  sub 
sequent  experience  has  removed  all  serious  objections  to  the  methods 
proposed  and  advocated  by  the  latter.  On  the  other  hand,  the  experi 
ence  of  Messrs.  Thiele  and  Zylinski  has  brought  out  in  stronger  light  a 
peculiar  personal  equation  that  had  been  long  known  to  exist  in  the  use 
of  meridian  transit  instruments  and  of  the  vertical  circle  for  time  deter 
minations.  It  is  found,  namely,  that  the  personal  equation  varies  (in  eye 
and  ear  observations  very  decidedly,  but  far  less  so  in  eye  and  hand  ob 
servations)  with  the  direction  of  the  motion  of  the  star's  image  over  the 
retina.  This. source  of  error  is  not  eliminated  by  the  reversion  of  the 
ordinary  direct- vision  transit  in  the  Y's,  but  may  be  so  if  a  reflecting 
eye-piece  be  properly  employed. 

On  account  of  the  great  saving  of  time  and  labor  afforded  by  the  use 
of  the  Brauer  transit  and  Dollen's  formulae,  these  especially  commend 
themselves  to  travelers  and  to  those  who  are  placed  in  unfavorable  at 
mospheric  conditions ;  the  extra-meridional  method  here  developed  must 
be  considered  indispensable  when  the  stability  of  the  transit  in  azimuth 
or  the  visibility  of  the  appropriate  stars  cannot  be  relied  upon  for  sev 
eral  hours,  but  may  be  for  perhaps  five  or  ten  minutes. 

It  will  often  be  found  highly  advantageous  for  the  observer  to  apply 
any  ordinary  portable  transit  to  observations  of  this  class  however  incon 
venient  the  instrumental  arrangements  may  seem  to  be. 

For  perspicuity  and  the  observer's  convenience  we  take  the  liberty  of 
appending  the  following  review  of  Dollen's  method,  together  with  a  few 
simple  tables. 


THE  VERTICAL  OF  THE  POLE  STAR.  33 

ORIENTATION. 

a-, 

Arrived  at  any  new  station  whose  latitude  as  well  as  our  clock^hro- 
nometer  correction  are  approximately  known,  we  determine,  first,  the 
zenith  point  of  the  setting  circle  by  observation  of  the  nadir  point  or  of 
a  distant  terrestrial  object.  Then  a  finding  ephemeris  (such  as  is  given 
in  the  appended  Table  I)  enables  us  to  set  a  ^the  proper  zenith  distance 
for  Polaris,  and  to  sweep  in  azimuth  until  that  star  is  found.  This,  if  it 
benight,  or  if  the  north  point  be  accurately  known,  will  require  but  a 
few  minutes. 

In  the  day-time  it  will  be  more  convenient  to  raise  the  telescope  by 
means  of  its"  sworvmg  apparatus  and  sight  upon  the  sun.  The  observed 
time  of  transit  (T)  and  the  zenith  distance  (z]  of  the  sun's  center  give  us 
his  azimuth  (A)  and  the  clock  correction  ( J  T)  by  the  following  well- 
known  fornmlse : 

Put 

90°  —  8  =  a  s  —  a  =  a 

900 ^l)  S &  =  /9 


where  the  double  computation  of  2  s  serves  as  a  check : 

Vsin  a  sin  ft  sin  Y 
; ' 
sins 


where  the  last  equation  again  serves  as  a  che.ck. 

Knowing  thus  the  sun's  azimuth,  A,  and  hour  angle,  t,  we  have 

North  point  =  observed  azimuth  i  A 
Clock  correction  =  T  —  (right  ascension  of  sun  +  t). 

The  ordinary  portable  transit  offers  no  very  convenient  means  for 
measuring  azimuths  or  for  turning  the  instrument  around  through  any 
given  azimuthal  angle.  In  the  Brauer  instrument  this  may  be  easily 
effected  by  means  of  a  simple  graduation  on  the  vertical  face  of  the  ex 
terior  rim  of  the  circular  base  of  the  transit,  or  more  elegantly  by  means 
of  a  graduation  in  the  circular  base  of  the  reversing  apparatus  or  the 
adjacent  bed-plate. 

If  the  instrument  does  not  contain  too  much  iron  in  the  construction 
of  its  parts  an  attached  compass  needle  will  be  found  convenient. 

Polaris  being  found  in  the  field  of  view,  the  careful  observer  will,  after 
leveling  the  axis,  reverse  upon  that  star,  or  upon  a  distant  terrestrial 
object,  and  see  that  the  collimation  error  is  not  too  large. 

The  choice  of  a  time  star  will,  in  the  day  time,  depend  upon  its  own 
brightness,  and  a  general  catalogue  of  all  stars,  to  the  third  magnitude 
inclusive,  will  afford  a  number  of  convenient  stars  at  any  time  or  in  any 
latitude.  We  have  appended  such  a  catalogue^ compiled  by  -  -  , 
including  all  stars  given,  as  of  the  third  or  brighter  magnitudes,  in  the 
British  Association  Catalogue,  or  Argelander's  Uranometria  Nova,  or 
in  Taylor's  General  Catalogue ;  to  these  stars  we  have,  however,  pre- 
3 


34          THE  PORTABLE  TRANSIT  INSTRUMENT  IN 

ferred  to  affix  Argelander's  magnitudes,  excepting  for  those  south  of 
—  25°  of  declination  j  for  the  variable  stars  we  have  adopted  the  limit 
ing  magnitudes  given  by  Schonfeld.  These  are  all  converted  into  the 
decimal  system  of  notation,  assuming  Argelander's  1.2  and  2.1,  &c.,  equal 
to  1.4  and  1.6,  &c.,  of  the  decimal  system.  For  night  observations  the 
brightness  of  the  star  is  not  always  a  matter  of  so  much  importance, 
though  it  will  be  conducive  to  greater  uniformity  of  personal  equation  if 
stars  of  equal  magnitude  be  habitually  employed.  On  the  other  hand, 
as  we  may  save  some  little  labor  by  using  the  stars  whose  apparent  places 
are  given  in  the  Annual  Astronomical  Ephemerides,  we  have  therefore 
added  to  our  list  of  200  bright  stars  all  others  given  in  the  American 
Astronomical  Ephemeris,  (A,)  in  the  British  Nautical  Almanac,  (B,)  in 
the  Berliner  Jahrbuch,  ( J,)  or  in  the  Coimaissance  des  Temps,  (0,)  and  have 
indicated  the  occurrence  of  a  star  in  either  of  these  almanacs  by  an  ap 
propriate  letter  in  the  second  column  of  the  resulting  catalogue  of  3G8 
stars. 

It  will  be  noticed  that  we  thus  obtain  a  sufficient  number  of  points 
for  use  in  observing  moon  culminations  should  the  traveler  have  time  or 
inclination  to  resort  to  that  method  of  determining  longitude. 

METHODS  OF  OBSERVATION. 

The  course  now  to  be  pursued  by  the  observer. depends,  to  some  extent, 
upon  the  manner  in  which  he  proposes  to  nse  his  instrument  for  the  ac 
curate  determination  of  his  clock  corrections,  but  chiefly  upon  the  atmo 
spheric  conditions  and  the  stars  of  his  ephemeris.  Either  of  the  follow 
ing  methods  may  be  adopted  ;  for  simplicity  we  shall,  as  usual,  speak  of 
the  observer  as  being  in  the  northern  hemisphere : 

I. 

The  instrument  being  moved  in  azimuth  until  Polaris  is  within  the 
reticule,  and  by  preference  'near  the  middle  thread,  we  read  the  level  in 
both  positions  of  the  axis,  observe  the  transit  of  Polaris  liear'oue  thread 
and  that  of  a  time  star  immediately  before  or  afterwards,  and  again  read 
the  level  in  both  positions.  Then,  supposing  the  inequality  of  the  pivots 
and  the  collimation  to  be  known,  we  have  the  material  for  determining 
the  clock  correction.  The  collimation  had  best  be  at  once  determined 
by  observations  of  Polaris  in  both  positions  of  the  axis. 

But  this  method  ought  never  to  be  resorted  to  except  in  extreme  neces 
sity  :  it  is  always  of  importance  to  institute  observations  of  the  time 
stars  as  well  as  of  Polaris  in  both  positions  of  the  axis,  as  in  the  follow 
ing  methods. 

IT. 

Eead  the  level ;  observe  the  time  star,  and  then  Polaris,  over  any 
thread,  by  preference  the  middle  one,  or,  still  better,  make  several  bisec 
tions  by  means  of  the  micrometer  thread;  read  the  level ;  reverse  the 
axis :  read  the  level ;  observe  Polaris  again,  as  also  another  time  star. 

The  observations  of  the  time  stars  may  either  precede  or  follow  those 
of  Polaris. 

This  method,  in  which  the  azimuth  remains  the  same  in  the  two  posi 
tions  of  the  axis,  is  specially  applicable  when  the  azimuthal  motion  of 
Polaris  is  rapid,  and  differs  from  the  ordinary  use  of  the  transit  only  in 
that  Polaris  is  observed  at  any  hour  angle  whatever. 


THE    VERTICAL    OF    THE    POLE    STAR.  35 

III. 

Read  the  level  ;  observe  Polaris  on  the  middle  thread  and  a  time  star; 
read  the  level  ;  reverse  the  axis  ;  read  the  level  ;  alter  the  azimuth 
slightly,  so  that  Polaris  will  again  cross  the  middle  thread  in  a  few  sec 
onds  ;  read  the  level  ;  observe  Polaris  on  the  middle  thread  and  a  time 
star;  read  the  level;  reverse  the  axis,  and  again  read  the  level. 

The  last  reversion  and  level  reading  may  be  omitted  if  we  have  no 
reason  to  doubt  the  stability  of  the  instrument. 

In  this  method,  which  is  commonly  the  most  convenient  and  expedi 
tions  both  as  regards  the  observations  and  the  computations,  we  alter 
the  azimuth  only  by  a  small  amount,  so  as  to  overtake  Polaris  in  its  di- 
nrnal  motion.  This  method  is  specially  applicable  when  no  micrometer 
is  provided  for  the  instrument. 

IY. 

By  the  azimuthal  motion  bring  the  extreme  western  thread  of  the  ret 
icule  to  Polaris  ;  read  the  level  ;  observe  the  time  star  and  Polaris  ;  read 
the  level  ;  reverse  the  axis  ;  read  the  level  ;  bring  the  same  thread,  now 
become  the  extreme  eastern  one,  to  Polaris  ;  read  the  level  ;  observe 
Polaris  and  the  same  time  star  ;  read  the  level  ;  reverse  the  axis,  and 
again  read  the  level. 

The  last  reversion  and  level-reading  may  be  omitted  if  the  mounting 
is  sufficiently  stable. 

The  fixed  thread  of  the  reticule  may  be  advantageously  replaced  by 
the  extreme  micrometer  thread,  if  such  an  one  is  provided  in  addition  to 
the  central  thread.  In  this  method  we  reverse  upon  Polaris  in  order  not 
to  alter  the  azimuth  so  much  as  to  leave  that  star  outside  the  reticule. 
It  is,  however,  more  expeditious  and  quite  safe  to  observe  Polaris  before 
the  time  star  ;  then  reverse  and  set  upon  the  same  time  star,  alter  the 
azimuth  so  as  to  again  observe  its  transit,  and,  finally,  observe  Polaris, 
which  will  certainly  be  found  in  the  field. 

In  this  method  the  azimuth  is  altered  by  a  large  amount,  not  exceed 
ing,  however,  the  angular  distance  of  the  extreme  wires  of  the  reticule. 

PREPARATORY  COMPUTATIONS. 


In  preparing  for  eifcta?  of  the  preceding  methods  of  observation  we 
make  use  of  a  solution  of  the  problem  giving  the  positions  of  two  stars 
and  of  the  observer's  zenith,  required  the  moment  at  which  these  three 
points  are  in  the  same  vertical  plane.  Instead  of  a  rigorous  solution, 
however,  we  may,  by  means  of  Table  II,  attain  a  sufficiently  approximate 
determination  of  the  required  moment. 

For  a  star  in  the  meridian  the  movement  in  azimuth  is,  approxi 
mately, 

,-  .  cos  3       T. 

d  A  =  —         -^r  d  t. 

sin  (tp  —  8) 

Table  II  gives  the  values  of  dA  for  dt  =  l;  its  use  will  be  apparent 
from  the  following  example: 

Example.—  At,  latitude  +  33°,  and  about  16h  of  sidereal  time,  we  desire 
to  determine  the  clock  correction  ;  we  select  d  Ophiuchi  and  Y  Herculis 
as  appropriate  time  stars,  whose  places  are 

/«.      m.  °        ' 

.  No.  234,  3  Ophiuchi  ;  3  ;  a  =  16     7.5  3  =  —   3     21 

No.  237,  Y  Herculis  ;  3  ;  a  =  10  10.2  d  =  +  19     28 


36 


THE  PORTABLE  TRANSIT  INSTRUMENT  IN 


For  these  two  moments  the  ephemeris  of  Polaris  gives 

h.     in.  /(.        m.  o 

At      =  330    atl°     7:      f  =  U     565      «  =  68     0;      A  =  l 
at  16  16:      t  =  I5      5:      z  =  5757;    "A  =  l 
From  Table  II  we  find 


5  =  181     5 

8  =  181     8 


f  For  d  Ophiuclii        =  +  1.8 

dA 
[  For  r  Herculis  ^==  +  2.2 

Whence  the  hour-angle  of  5  Ophiuchi,  when  at  the  azimuth,  —  1°  5',  is 

0°.GO  =  —  2m.4 


10  y 
=r  ?- 

1.0 


and  for  r  Herculis,  £  =  —  -^  _0°.51  =  —  2m.l 

and  the  moments  at  which  the  time  stars  are  to  be  observed  will  be, 
respectively,  16h  9m.9  and  16h  18m.3. 

In  setting  for  the  time  stars  we  may  use  meridian  zenith  distances, 
<p  —  d,  but  having  brought  the  star  between  the  horizontal  wires,  when 
once  it  has  entered  the  field,  we  should,  for  accuracy  and  security,  record 
the  reading  of  the  setting  circle,  and  that,  too,  if  possible,  to  the  fraction 
of  a  minute. 

In  following  method^  IV  of  observation  we  need  to  know  the  space  of 
time  within  which  the  observation,  reversal,  &c.,  must  be  accomplished, 
or  the  change  of  azimuth  that  can  be  made  without  throwing  Polaris 
too  far  from  the  middle  wire. 

We  have,  with  sufficient  accuracy, 


sin  z 


where  i  =  the  equatorial  distance  of  the  extreme  from  the  middle  thread, 
and  z  is  the  zenith  distance  of  the  time  star  to  be  observed.  Assuming 
i  =  lm  =  15',  we  obtain  the  d  A  of  the  following  tables : 


z. 

6  A. 

0 

0 

/ 

10 

1 

26 

15 

0 

58 

20 

0 

44 

25 

0 

36 

30 

0 

30 

z. 

(5-  A. 

o 
30 

o 
0 

30 

40 

0 

23 

50 

0 

20 

60 

0 

18 

70 

0 

16 

The  azinmthal  change  to  be  given  to  the  transit  is,  evidently,  double 
the  A  of  the  preceding  table,  since  the  same  wire  is  to  be  observed  first 
wrhen  on  the  extreme  west,  and  again  when  on  the  extreme  east. 

Example. — At  latitude  54°,  at  about  12h  35m  sidereal  time,  we  propose 
to  observe  f  Yirginis  for  clock  correction.     We  have : 
Polaris          z  =  —  37°  22'      A  =  +  0°  19'       t  =  llh  24m       d  A  =  0°  25' 

fj  A 

r  Virginia      z= 


THE    VERTICAL    OF    THE    POLE    STAR.  37 

Whence,  if  lm  is  the  equatorial  distance  of  the  extreme  wire,  the  two 
setting's  must  be  between  the  azimuths 

AI  =  +  0°  19'  +  0°  25'  =  +  Oo  44/?  aiuj  A2  =  +  0°  19'  —  (P  25'  =  —  0°  6'  ; 

and  the  time  elapsed  between  the  time  star  transits  over  the  middle 
threads  of  the  diaphragm  will  be 

QQ50/_31".33 
~L2T  =  =  1.1M 

leaving  thus  at  least  forty  seconds  for  reading  the  level  and  setting  cir 
cle,  reversing,  reading  the  level,  and  setting  at  the  proper  zenith  distance 
and  azimuth.  This  is,  indeed,  as  unfavorable  a  case  as  need  occur.  By 
omitting  to  observe  the  time  star  on  the  extreme  wires,  we  can  mate 
rially  increase  the  interval  of  forty  seconds.  In  instruments  properly 
constructed  the  level  is  always*  in  place  and  its  reading  occupies  but  a 
few  seconds. 

METHODS  OF  COMPUTATION.  —  COLLIMATION. 

In  case  the  collimation  is  to  be  independently  determined  by  observa 
tions  of  Polaris  in  reversed  positions  of  the  axis,  we  may  use  the  follow 
ing  method  for  deducing  the  value  of  that  quantity  : 

The  several  micrometric  bisections  being  made  in  rapid  succession  in 
each  position  of  the  axis,  it  will  be  sufficiently  exact  to  consider  the  mean 
reading  of  the  micrometer  as  belonging  to  the  mean  of  the  observed 
times.  We  have,  then,  as  also  in  case  any  two  fixed  wires  are  observed, 

At  time  T1?  first  position,  Polaris  distant  from  the  middle  wire  -  -  ii 
At  time  T2,  reversed  position,  Polaris  distant  from  the  middle  wire  -  i2 

the  collimation  c,  and  the  distances  ii  and  i2,  being  always  supposed  to 
increase  in  the  direction  of  the  positive  motion  of  the  micrometer  screw 
and  to  be  expressed  in  seconds  of  time,  being  known  by  observations  of 
equatorial  and  polar  stars. 
Now,  the  ordinary  formula  for  azimuth  instruments  give, 

At  time  T1?  AI  =  «i  -f  &i  cotg  Zx  +  (c  +  ii)  cosec  Zi, 
At  time  T2,  A2  =  «2  -f  &2  cotg  Z2  -f  (c  -f  i2)  cosec  Z2. 

Where  A!  and  A2  are  the  azimuths,  Z1?  Z2  the  zenith  distances  of  Po 
laris,  Oi  =  &0  4-  3&P  and  «2  =  «•<>  +  (^aP  the  azimuths  of  the  axis  of  revo 
lution  in  the  direct  and  reversed  positions,  and  which  will  be  identical 
if  the  pivots  are  of  perfect  form,  (or  oa  p  =  0  and  3fap  =  0.)  So,  also,  will 
the  inclinations^^  b0  -f  J/^>  4-  4%p  and  b2  =  b0  -f  A\p  +  A'2p  be  identi 
cal  if  the  pivots  are  of  equal  diameters  and  perfect  form,  (or  Al  p  =  0  ; 
J'!|)  =  0;  Ja#  =  0;  ^^  =  0.) 

For  Zl  and  Z2  we  may  use  the  mean  zenith  distance,  Z,  taken  from  the 
finder,  or  the  finding  epheineris  for  the  mean  moment,  J  (Ti  -f  T2)  =  T. 

As  the  observations  are  supposed  to  be  made  in  rapid  succession,  we 
ma  assume 


— 


where 


38  THE    PORTABLE    TRANSIT    INSTRUMENT    IN 

Q  being  the  parallactic  angle,  which  is  given  with  sufficient  accuracy  by 

sinQ  =  cos^t-^. 
sinZ 

There  now  results,  for  the  collimatiou, 

*!tTq^^ 


c  will  be  negative  if  the  middle  thread  or  micrometer  zero  is  too  far  in 
the  direction  of  positive  motion. 

TIME. 

The  deduction  of  the  clock  correction  js  to  be  effected  by  slightly  dif 
ferent  processes  of  computation,  according  as  we  have  followed  one  or 
the  other  of  the  previous  methods  of  observation.  The  formulae-  given 
by  Dollen  may  be  systematically  arranged  as  follows,  using  the  follow 
ing  notation  as  adopted  from  the  preceding  memoir  : 

Pole  Star.     Time  Star. 

S'  S         the  observed  times  of  transit. 

u'  =  U-\-Y      u          the  clock  correction^  on  sidereal  time  ;  y  the  correc 

tion  for  clock  rate  during  S'  —  S. 
a'  a          the  apparent  right  ascensions,  corrected  for  diurnal 

aberration,  if  necessary. 
d'  d          -the  apparent  declinations. 

z'=o'—y  z=y—<l   the  zenith  distances,  z'  being  always  positive. 

/  the  interval  for  the  thread  on  which  Polaris  is  ob 

served,  (+  when  west  of  the  middle.) 
c  the  error  of  collimation,  (positive  when  the  middle 

thread  lies  too  far  to  the  west.) 

1)  the  inclination,  (positive  for  west  end  high.) 

a  the  azim.uth  of  the  axis,  (positive  when  its  west  end 

lies  south  of  west  point.) 

COLLIMATION  KNOWN. 

One  pole  star  and  one  time  star,  observed  in  either  position  of  the 
axis;  the  inclination,  (&,)  the  latitude,  (&,)  and  thread  interval,  (/,) 
known. 

1.  —  Rigorous  -solutioy. 

(See  Dollen,  §  14,  first  solution.) 
(«.)  Compute  r  from 

S'  +  r  —  a'  =  D';     S  —  a  =  D;     15(D'  —  D)  =  r. 
<L/     (6.)  Compute  sin  |c,  cos  d,  and  £  from 

1  Ji,/  cos  /  sin  c  =  cos  d'  sin  r 

cos  d  cos  c  =  cos  d  sin  d'  —  sin  d  cos  d'  cos  r 
sin  d         =  sin  8  sin  d'  +  cos  d  cos  d'  cos  r. 
(c.)  Compute  >?  from 

sin  (c  +/)  —  sin  c  sin  d 
sin  79  =  —  —  . 

cose  cos  d 


THE    VERTICAL    OF    THE    POLE    STAR.  39 

(d.)  Compute  n,  x,  and  m  from 

COSMCOS#  =  +  cose  cos  (£  4-  r;) 
cos  n  sin  x  —  —  cos  3  sin  c  -f  sin  d  cos  c  sin  (*  -f-  rt) 
sin  n  =  -f  sin  (5  sin  c  +  cos  5  cos  c  sin  (£  +'  -//) 

sin  w         =  tg  n  tg  9?  +  sin  h  sec  ?i  sec  ^ 

and  deduce  ^  (a?  —  m). 

JLO 

(e.)  Eeduce  each  observed  thread  to  the  middle  one  by  the  reductions 

~~tr/       ~*&-=f  Vsec  (d  +  n)  sec  (d  —  n) 

and  derive  a  new  S  from  the  mean  of  all,  whence  a  new  D  is  to  be  found. 
This  will  rarely  necessitate  a  correction  to  the  ^  (x—m)  already  com-' 

puted. 

(/.)  With  the  corrected  D  compute 

u  =  ^~(x  —  m)  —  D 

2.  —  Approximate  solution  . 

•When  F/  is  large,  or  the  factor  F  is  not  given.     (See  Dollen,  §  14, 
third  solution.) 

(a.)  Compute  r  as  in  (1  .  a)  from 

g/  +  r_a/  =  D';     S  —  «  =  D;     15(D'  —  D)  =  r. 

(6.)  Compute  £  from 

sec  £  cotg  <5X  sin  r 

4-  0<  £:  ___  _____  o  ______  ^ 

"1  —  tg  5  COtg  d'  COS  r 

(c.)  Compute  -/?  from 

sin/" 

8111  *  =  tin  (*  +  *)' 

(d.)  Compute  x\  and  ?«!  from 


Sin  Wi  =  COS  ^  tg  (I  +  ry)  COS  iPi  tg  ^. 

(e.)  We  now  have 


15  v 

and  there  remains  only  to  reduce  all  the  transits  of  the  time  star  to  the 
middle  thread,  to  correct  D  thereby,  and  to  compute  the  terms Bb  +  Cc. 

If  the  value  of  S,  first  used,  be  the  mean  of  all  threads  systematically 
disposed  about  the  mean  thread,  it  will,  generally,  not  require  any  cor 
rection  for  the  quantity^  n. 

(f.)  The  inclination,  o,  is  given  in  level  divisions,  one  of  which  has 
the  value,  _p,  expressed  in  seconds  of  time  ;  whence  we  can  easily  com 
pute 

B 1)  =  l)p  sec  (p. 

(g.)  The  collimation,  c,  is  expressed  in  seconds  of  time,  and  we  have  to 
compute 

C0  =  sec  <f>  cos  J  (z1  —  z)  sec  £  (z1  -f  z)  *.   ^     > 

q  =  cosec2 <p  +  (cotg <p  —  tg jrttg J (z1  —  z)  f  f / 

C  =  C0  (sec  mi)  i;  '/^\ 

whence  results      C .  c. 


40  THE    PORTABLE    TRANSIT    INSTRUMENT    IN 

It  will,  if  c  be  moderately  small,  be  always  allowable  to  use  C0  instead 
ofC. 

3. — Approximate  solution. 

When  F/is  not  large  or  the  factor  F  is  given.     (See  Dollen,  §  14,  sec 
ond  solution.) 

(a.)  Compute  r,  as  before,  from 

S'  +  r —  a'==D';     S  —  a  =  D-     15  (D'  —  D)  =  r; 

where  we  shall  not  materially  increase  the  probable  error  of  our  result 
by  adding  to  the  threads,  observed  symmetrically,  any  others  reduced  to 
the  mean  thread,  by  the  ordinary  formula, 

t     (&.)  Compute  A,  fjL7  v,  and  />,  from 

A  =  tg  d  cotg  3' 

si  n  ""* 

/ti«=tg?eotg<5  />  =  i-  -• 


v  =  A  //  =  tg  c?  COtg  d1 

(c.)  Compute  x0  and  MO  from 

tg  x0  —  '>  />,  sin  M0  =  v  p  cos  ,r0. 
rf.    We  now  have 


and  it  remains  to  compute  the  three  terms, 

B6  +  Cc+F/. 

(e.)  As  in  (2  ./)  we  have,  for  the  inclination, 


(/.)  For  the  collimation  we  have,  as  in  (U  .  //), 

sin  ^^  --I--  mil  2j 
°°  =  S6C  *  ~~=  S 


=  cosec 


+  f^  1  -      jcotg  ^  tg  J  (^x  — 


and  C  = 

where,  as  before,  for  all  ordinary  values  of  c,  we  may  assume  C  equal  to 
C0. 

(g.)  The  equatorial  thread,  or  micrometer  interval,  /,  being  expressed 
in  seconds  of  time,  we  have 


sin  (z1  +  z) 


and  F  ==  F0  (sec  m0)k. 

If  F  is  not  given  by  appropriate  tables,  then,  for  very  small  values  of 
/,  we  may  assume  F  equal  to  F0. 

COLLIMATION   UNKNOWN. 

Each  of  the  three  methods  of  computation  previously  given  assumes 
the  collimation  (c)  to  be  previously  known,  and  is  adapted  to  the  inde 
pendent  deduction  of  a  clock  correction  for  each  pair  of  stars  observed 


THE  VERTICAL  OF  THE  POLE  STAR.  41 

in  either  position  of  the  instrument.  But  it  is  evident  that  we  have  only 
to  make  a  very  slight  change  in  the  order  of  computation  so  as  to  deduce 
both  colliraation  and  clock  correction  from  the  observation  of  two  pairs 
of  stars,  one  in  each  position.  It  is  this  class  of  observations  that  is  to 
be  highly  recommended  and  especially  enjoined. 

We  may  leave  out  of  consideration  the  rigorous  methods  of  computa 
tion  and  confine  ourselves  to  the  more  generally  useful  approximate 
methods. 

4.  —  Approximate  solution. 

When  F/is  large,  or  the  factor  F  not  given.  (See  Dollen,  §  15,  first 
example.) 

The  azimuth  may  have  remained  the  same  and  two  different  stars  have 
been  used  for  time  stars,  (as  in  the  II  method  of  observation,)  or  the  azi 
muth  may  have  been  altered  in  order  to  observe  the  same  time  star  in 
both  positions,  (as  in  the  IY  method  of  observation.)  In  either  case  we 
have  S'i  and  SL  for  the  observations  of  the  Pole  Star  and  Time  Star  in  the 
first  position,  but  S'2  and  S2  for  the  observations  in  the  reversed  posi 
tion,  and  we  have  to  compute  the  observations  of  each  position  inde 
pendently,  according  to  the  method  2  merely  in  article  (2.0),  stopping 
at  the  computation  of 

C0,  #,  and  C. 

We  thus  obtain,  by  (2  .  e),  from  the  first  time  star, 

w1±C1c=115(a?1  —  i»i)  —  (D.+  B^), 
and  from  the  second  time  star, 


Since  each  clock  correction,  Ui  and  u^  is  supposed  to  hold  for  the  mo 
ment  of  the  observation  of  the  time  star,  we  reduce  them  (by  adding  the 
quantities  YI  and  -^  computed  with  an  approximate  clock  rate)  to  a  com 
mon  moment,  for  which  let  u  be  the  clock  correction  ;  we  then  have 


where  the  upper  signs  hold  when  the  first  position  of  the  instrument  is 
the  normal  one.  The  half  sum  and  difference  of  Ox  and  O2  now  give  us 
the  complete  solution  of  our  problem. 

5.  —  Approximate  solution. 

When  F/is  small,  or  F  previously  given.  (See  Dollen,  §  15,  second 
example.) 

'  The  azimuth  may  have  remained  the  same  and  Polaris  have  been 
observed  over  side  threads  or  the  micrometer  wire,  (as  in  the  II  method 
of  observation,)  or  the  azimuth  may  have  been  changed  and  Polaris  ob 
served  on  the  middle  thread,  (as  in  the  III  method  of  observation.) 

We,  in  the  first  case,  follow  the  method  3  with  a  modification  precisely 
similar  to  that  given  in  the  previous  method,  4,  and  we  obtain,  as  there, 
from  the  values  of  1/1  +  n  ±  do  and  of  u  +  r'2  ±  C2c  the  correct  u  and  c. 
(See  Dollen,  §  15,  first  example.) 

In  the  second  case,  following  method  III,  F/is  zero  ;  the  computation 


42  THE    PORTABLE    TRANSIT    INSTRUMENT   IN 

is  made  according  to  2  and  4,  but  becomes  simplified  since  TJ  =0.  This 
is,  probably,  the  best  of  all  the  methods  of  observation  and  computation. 
(See  Dollen,  §  15,  second  example.) 

NOTE  1.— If  we  have  pursued  the  IV  method  of  observation,  we  may  assume  Ci  — C2 
without  sensible  error  and  obtain  it  without  computing  C  ore.  But  it  is  always  desira 
ble  not  to  omit  the  determination  of  the  collimation. 

NOTE  2. — We  have  now  reviewed  the  five  methods  of  computation,  some  one  of  which 
is  specially  appropriate  to  the  use  of  the  transit  instrument^accordiug  to  either  of  the 
four  methods  of  observing  previously  given.  The  expert  observer  will  not  find  it  diffi 
cult  so  to  arrange  his  observations  that  five-place  logarithms  may  be  used  in  the  com- 
nutations,  and,  as  in  the  use  of  small  angles  the  ordinary  five-figure  tables  of  logar- 
ithmic-trigonometric  footers  may  be  advantageously  replaced  by  the  tables  of  logar 
ithms  of  numbers,  we  have  reproduced,  in  Table  III,  the  numbers  alluded  to  by  Dollen, 
as  given  by  WwwtgeJl,  by  means  of  Avhich  one  easily  passes  from  the  logarithm  of  any 
number  of  seconds  of  arc  to  the  corresponding  logarithmic  sine  or  tangent.  If  there 
be  required  the  logarithmic  sine  or  tangent  of  the  angle  x,  which,  expressed  in  seconds, 
we  denote  by  x",  then  we  have 

Briggiau  log  sine  x        —  Briggiau  log  x1'  -f-  4.685575  —  S 
Briggian  log  tangent  x  =  Briggiau  log  x"  -f-  4.685575  -f  T. 

Our  Table  III  gives  the  numbers  S  and  T  in  units  of  the  sixth  place  of  decimals.  It 
will  be  most  convenient  to  omit  the  addition  of  the  constant  logarithm  4.685575,  in 
cases  where  it  is  eventually  eliminated  from  the  result.  In  using  six-figure  logarithms 
the  computer  will  find  it  advantageous  to  copy  into  Bremiker's  six-figure  tables  the 
numbers  S  and  T,  given  in  the  Avell-known  edition  of  Vega  by  the  same  author. 

The  conversion  of  arc  into  time,  and  vice  versa,  should  be  avoided,  if  possible,  by  the 
use  of  tables  in  which  the  arguments  are  given,  in  parallel  columns,  both  in  time  and 
arc. 


THE    VERTICAL    OF    THE    POLE    STAR. 
GENERAL  STAR  CATALOGUE. 


43 


No. 

A.   B.   J.   C. 

Name. 

Mag. 

a  1870.0 

Annual 
variation. 

(51870.0. 

Annual 
variation. 

ft.  m.  s. 

g 

o     /      n 

„ 

1 

A    B    J    C 

a    Andromedae  . 

2 

0    1  40 

+       3.1 

+  28  22  22 

-       19.9 

2 

0    Cassiopea3           .     . 

2.4 

0    2  13 

+       3.1 

+  58  26    0 

+       19.  9 

3 

A    B    J    C 

y    Pegasi    ,  . 

2.6 

0    6  33 

+       3.1 

+  14  27  39 

-       20.  0 

4 

A    B 

0    Hydrw  u  .     .     .     . 

3 

0  18  53 

+       3.3 

—  77  59  14 

J-      20.2 

5 

C 

a    Phoenicia      .     .     . 

2 

0  19  50 

+       3.0 

—43    0  38 

J-       19.  7 

6 

B 

12  Ceti     

6 

0  23  24 

+       3.1 

—    4  49  33 

+       19.  9 

7 

6    Andromedae  . 

3.4 

0  32  23 

+       3.2 

+30    8  55 

19.7 

8 

A    B    J    C 

a    CassiopeaB     .     .     . 

2.  2-2.  8 

0  33    9 

+       3.4 

+  55  49  26 

+       19.8 

9 

A    B    J 

0    Ceti           .... 

2 

0  37    4 

-f       3.0 

—  18  42    2 

-       19.8 

10 

A 

21  Cassiopea; 

6 

0  37    6 

+      3.8 

+  74  16  34 

19.7 

11 

y    Cassiopege     .     . 

2 

0  48  53 

+       3.5  ' 

+60    0  44 

-       19.6 

12 

A    B          C 

£     Piscium  .... 

4 

0  56  12 

+       3.1 

+     7  11  22 

19.5 

13 

rj    Ceti      

3 

123 

+       3.0 

—  10  52  11 

19.2 

14 

C 

0    Andromeda?  .     .     . 

2.4 

1    2  28 

+       3.3 

+  34  55  53 

+       19.  3 

15 

A    B    J    C 

a    Ursc'e  Miuoris    .     . 

2 

1  11  16 

+     20.1 

+  88  36  59 

-       19.1 

16 
17 

A    B 

8    Cassiopeee 
9    Ceti     

3 
3 

1  17  21 
1  17  31 

+       3.8 
+       3.0 

+  59  33  30 
—    8  51  18 

+       18.9 
+       18.7 

18 
19 

A 

A  (38)  Cassiopere    .     . 
y    Phomicis       .     .     . 

6 
3 

1  21  36 
1  22  44 

+       4.4 
+      2.6 

+  69  35  39 
—  43  59    2 

18.7 
r        18.  6 

20 

A    B 

TI    Piscium    .... 

3.6 

1  24  32 

+      3.  2 

+   14  40  30 

+       18.7 

21 

A    B          C 

a    Eridani    .... 

1 

1  32  52 

+      2.2 

—  57  53  51 

+       18.4 

22 

B 

v    Piscium    .... 

4.6 

1  34  40 

+       3.1 

+     4  49  43 

+       18.3 

23, 

C 

0    (54)  Aiidromeda3     . 

4.  4 

1  35  32 

+       3.7 

+  49  49  44 

+       18.  4 

.24' 

A 

o    Piscium   .... 

4 

1  38  32 

+      3.2 

+     8  30    8 

+       18.  3 

25 

g    Ceti     

3 

1  45    3 

r       3.0 

—  10  58  45 

+       17.9 

26 

£     Cassiopea)     . 

3.4 

1  45    4 

+       4.2 

+  63    1  44 

-4-      18.0 

27 

A    B          C 

0    Arietis          .     .     . 

2.6 

1  47  28 

+       3.3 

+  20  10  17 

-     -17.8 

28 

A 

50  Cassiopea) 

4 

1  52  23 

+       5.0 

+  71  47  24 

-       17.7 

29 

a    Hydrss.  JV 

3 

1  54  40 

+      1.9 

—  62  12    2 

-       17.  6 

30 

y    Andromedae 

2.4 

1  55  55 

+      3.6 

+  48  17  40 

-       17.6 

31 

A    B    J    C 

a    Arietis     .... 

2 

1  59  51 

+      3.4 

+  22  50  48 

+       17.  2 

32 

0    Trianguli      .     .     . 

3 

2    1  49 

+       3.5 

+  34  12  28 

-       17.3 

33 

A 

£l    (65)  Ceti  .... 

4.  4 

267 

+       3.2 

+     8  14    8 

17.1 

34 

B 

67  Ceti     

6 

2  10  30 

+       3.0 

—    71  21 

+       16.  7 

35 

c 

o    Ceti 

1.  7-OO 

2  12  47 

+       3.0 

—    3  34  10 

+       16.6 

36 
37 

A                C 
B 

i     (35)  Cassiopea?  .     . 
f2  Ceti     

4 
4 

2  18  23 
2  21  15 

+       4.8 
+       3.2 

+  66  48  56 
+    .7  52  33 

J-       16.5 
+       16.3 

33 

A    B    J 

y    Ceti     

3.4 

2  36  34 

+       3.1 

+     2  41  10 

-       15.4 

39 

C 

41  Arietis     .... 

4 

2  42  20 

+       3.5 

+  26  43  26 

-      15.2 

40 

77    Eridani    .... 

3 

2  50     4 

+      2.9 

—    9  25     1 

~      14.6 

41 

y    Persei      .... 

3 

2  55  24 

+       4.3 

+  52  59  42 

-      14.  6 

42 

A    B    J    C 

a    Ceti     

2.4 

2  55  29 

+       3.1 

+     3  34  41 

14.  4 

43 

C 

0   Persei       .... 

2.  S...4.  0 

2  59  43 

+       3.9 

+  40  27  10 

+       14.  3 

44 

A 

48  Cephei      .... 

6 

3    3  55 

+       7.3 

+  77  15    9 

-        13.  9 

45 

B    J 

i    Arietis     .... 

4.4 

3     4  12 

+       3.4 

+  19  14    0 

-       13.9 

46 

A 

£    Arietis     .... 

4.4 

3    7  26 

+      3.4 

+  20  33  39 

13.7 

47 

A    B    J    C 

a    Persei       .... 

2 

3  15    3 

+       4.2 

+  49  23  45 

-      13.  2 

48 

£     Eridani    .... 

3 

3  26  45 

+      2.9 

—    9  54    3 

+      12.4 

49 

A                 C 

J    Persei       .... 

3 

3  33  44 

+       4.2 

+  47  22    9 

-       11.9 

50 

J    Eridani    .... 

3 

3  37    2 

+       2.9 

—  10  12  20 

+       10.0 

51 

A    B 

rj    Tauri  .... 

3 

3  39  46 

+      3.6 

+  23  42    3 

+       11.5 

52 

A 

g    Persei            .     .     . 

3 

3  45  58 

+       3.8 

+  31  29  42 

-i-       11.0 

53 
54 

A    B 

y    Hydm*/    .    .    . 

y    Eridani  /  .     .      .     . 

3 
3 

3  49  18 
3  51  58 

—      1.0 

+      2.8 

—  74  38  16 
—  13  52  49 

-      10.9 

4-      10.5 

55 

B 

o1    Eridani    .... 

4.4 

4     5  31 

+       29 

—    7  10  42 

+        9.7 

56 

A                 C 

y    Tauri  ...... 

4 

4  12  24 

+       3.4 

+  15  18  41 

+        9.1 

57 

A    B 

£     Tauri  

3.6 

4  21    3 

+       3.5 

+  18  53  22 

+        8.4 

58 

A    B    J    C 

a     Tauri  

1 

4  28  28 

+       3.4 

+  16  14  45 

+        7.6 

59 

52  Eridani     .... 

3      Ctf 

'   4  29-30 

+       2.3 

—  30  49  47 

+         7.7 

60 

a    Doradus  .... 

3      // 

4  31  11 

+       1.3 

—  55  13  53 

4-         7.6 

61 
62 

A 
C 

a    (9)    Camelopardalis 
Tr1   Orionis     .... 

4 
5 

4  41     8 
4  42  47 

+       5.9 
+       3.3 

+  66    7    4 
+     6  43  56 

-r          6.8 
6.7 

63 

A    B 

t     Aurigae    .... 

3 

4  48  32 

+       3.9 

+  32  57  27 

-r         6.1 

64 
65 

C 
A 

0   (10)  Camelopardalis 
11  Orionis     .... 

4 
5 

4  51  52 
4  57    9 

+       5.3 
+       3.4 

+  60  14  55 
+  15  13  14 

+        6.0 
~        5.4 

44 


THE    PORTABLE    TRANSIT    INSTRUMENT    IN 
GENERAL  STAR  CATALOGUE— Continued. 


Xo. 

A.   B.   J.   C. 

Nam*. 

Mag. 

a  1870.0. 

Annual 
variation. 

<J  1870.0. 

Annual 
variation. 

h.  m.  s. 

s. 

o     /     // 

„ 

66 

B 

e     Leporis    .... 

3.6 

4  59  57 

4-      2.5 

—  22  32  51 

4-        5.1 

67 

ft    Eridani    .... 

3 

5    1  28 

+      2.9 

—    5  15  25 

-r        5.0 

66 

A    B    J    C 

d     A  11  ri  gffi 

1 

575 

4-       4.  4 

4-  45  51  45 

4-        4.2 

69 

A    B    J    C 

0    Oriouis      . 

1 

5    8  17 

4-       2.9 

—    8  21  15 

4-        4.5 

70 

A    B    J    C 

0    Tauri  

2 

5  18     5 

+       3.8 

4-  28  29  41 

4-        3.5 

71 

C 

y    Orionis     .... 

2 

5  18  10 

+       3.2 

4-     6  13  49 

4-         3.7 

72 

A 

966  Groom  bridge     .     . 

6.4 

5  22  22 

4-       8.0 

4-  74  57    5 

4-        3.3 

73 

A    B           C 

(5    Orionis     .... 

2.  2...  2.  7 

5  25  22 

4-      3.1 

—    0  23  53 

4-        3.0 

74 

A    B 

a    Leporis    .... 

3 

5  27     0 

+       2.6 

—  17  55    3 

4-        2.9 

75 

i     Orionis     .... 

3 

5  29     4 

4-       2.9 

—    5  59  50 

4-        2.8 

76 

A    B          C 

c     Orionis     .... 

2 

5  29  37 

+      3.0 

—    1  17  14 

4-        2.6 

77 

C 

g    Orionis     .... 

2 

5  34  12 

4-      3.0 

—    2    0  49 

+         2.3 

76 

A    B          C 

a    ColamlNB      .     .     . 

2 

5  34  57 

+      2.2 

—34     8  40 

4-        2.2 

79 

K    Orionis     .... 

2.6 

5  41  35 

+      2.8 

—    9  43    5 

4-        1.7 

80 

0    Columbse       .     .     . 

3 

5  46  23 

4-      2.1 

—  35  49  14 

4-         1.5 

81 

A    B    J    C 

a    Orionia     .... 

1.0..  .1.4 

5  48    8 

-f       3.2 

4-     7  22  49 

-f        1.1 

82 

0    AurigsB    .... 

2 

5  49     0 

4-      4.4 

4-  44  55  52 

+        1.0 

83 

0    Aurig»    .... 

3- 

5  50  51 

+      4.1 

4-  37  12    4 

4-        0.8 

84 

B 

v    Orionis     .... 

4.6 

609 

4-       3.4 

4-  14  46  53 

0  0 

85 

A 

22  Camelopardalis 

4.6 

6     4  31 

4-       6.6 

4-  69  21  36 

—        0.5 

86 

A    B 

p    G-eminorum  . 

3 

6  15     6 

+      3.6 

4-  22  34  38 

—        1.4 

87 

g    Canis  Majoris    . 

2.6 

6  15  20 

+      2.3 

—  30    0  28 

—        1.3 

88 

C 

0    Canis  Majoris    .     . 

2.4 

6  16  58 

4-      2.6 

—  17  53  36 

—        1.4 

89 

A    B          C 

a    Argus      .... 

1 

6  21     4 

4-       1.3 

—  52  37  32 

—        1.8 

90 

A    B 

y    Geminorum  .     .     . 

2.4 

6  30  12 

+      3.5 

4-  16  30  28 

—        2.7 

91 

v    Argus       .... 

3 

6  33  47 

4-       1.8 

—43    4  57 

—        2.9 

92. 

IS   Geminorum  .     .     . 

3.4 

6  35  59 

4-      3.7 

+  25  15  26 

—        3.0 

93 

A    B 

51  Cephei      .-..«. 

5 

6  38  42 

4-     30.3 

4-  87  14  23 

—        3.4 

94 

A    B    J    C 

a    Canis  Majoris    .     . 

1 

6  39  26 

f       2.6 

—  16  32  22 

—        4.6 

95 

A    B 

£     Cauis  Majoris    .     . 

1.6 

6  53  31 

+       2.4 

—  28  47  50 

—        4.6 

96 

B 

y    Canis  Majoris    . 

4.4 

6  57  53 

4-       2.7 

—  15  26  35 

—        5.0 

97 

A 

6    Canis  Majoris    .     . 

2 

736 

4-       2.  4 

—  26  11  18 

—        5.2 

96 

A    B 

6    Geminorum  . 

3.4 

7  12  21 

4-       3.6 

4-  22  13    8 

—        6.2 

99 

IT     Al'gUS          .... 

3.4 

7  12  34 

+      2.1 

—  36  51  55 

—        6.2 

100 

A 

Piazzi  VII,  67    .     . 

6 

7  17  20 

4-       6.3 

4-  68  43  35 

—        6.7 

101 

?7    Canis  Majoris    . 

2.6 

7  18  57 

4-       2.4 

—  29     3    4 

—        6.7 

1C2 

C 

0    Canis  Minoris    .     . 

3 

7  20     4 

4-      3.3 

4-     8  32  58 

—        6.8 

103 

A     B    J  "C 

a1  Geminorum  .     .      . 

1.6 

7  -20  Id 

4-       3.8 

4-  32  10  15 

—        7.4 

104 

A    B    J    C 

o    Canis  Minoris    .     . 

1 

7  32  30 

4-       3.1 

4-     5  33  22 

—        8.9 

105 

A    B    J    C 

0    Geminorum  .     .     . 

1.4 

7  37  22 

4-       3.7 

4-  28  20  16 

—        8.3 

106 

C 

£    Argus       .... 

3.4 

7  43  50 

+       2.5 

—  24  32    7 

—        8.7 

107 

A 

0    Geminorum  . 

5 

7  45  32 

4-       3.7 

+  27    5  58 

—        8.9 

108 

%    Argus      .... 

3 

7  53  28 

4-       1.5 

—  52  38    4 

—        9.5 

109 

B 

TO.   Caucri      .... 

5 

7  55  32 

4-       3.7 

4-28    9  23 

—        9.8 

110 

£    Argus      .... 

2.5 

7  59     1 

4-      2.1 

—  39  38  20 

—      10.0 

111 

A               C{ 

55  Camelopardalis       ^ 
3    Ursre  Majoris     .      5 

6 

j     7  59  51 

4-      6.1 

+  68  51  10 

—      10.0 

119 

113 

A    B 
C 

T\(15J;  Argus    .      .     . 
y2  Argus       .... 

3 

2 

820 
8    5  31 

4-      2.6 

4-       1.8 

—  23  55  52 
—  46  57  19 

—      10.1 
—      10.5 

114 

C 

0    Cancri      .... 

3.6 

8    9  27 

4-       3.3 

4-     9  35     3 

—      10.7 

115 

e    Argus       .... 

2 

8  19  51 

4-       1.2 

—  59    5  27 

—      11.3 

116 

B 

i?    Cancri      .... 

6 

8  25  11 

4-       3.5 

4-  20  52  50 

—      11.9 

117 

C 

f    Hydros      .... 

4.4 

8  30  46 

4-       3.2 

4-     6    9  21 

—      12.2 

118 

A    B 

£    Hydrae      .... 

3.4 

8  39  53 

4-      3.2 

+     6  53  39 

—      12.  9 

119 

<5    Argus       .... 

3 

8  41     7 

—      1.7 

—  54  14    0 

—      13.1 

120 

A    B    J 

i     Ursae  Majoris    . 

3 

8  50  18 

4-       4.1 

4-  48  33    0 

—      13.8 

121 

A 

<r*  Ursa?  Majoris    .     . 

5 

8  58  55 

4-       5.4 

4-  67  39  32 

—      14.2 

122 

A 

K    Cancri       .... 

5 

9     0  42 

4-      3.3 

4-  11  11  22 

—      14.2 

123 

e/ 

£    Cancri      .... 

5 

9     1  53 

4-      3.5 

4-  22  34  13 

—      14.2 

124 

A    Argus       .... 

3 

9     3  13 

4-       2.2 

—  42  54  33 

—       14.4 

125 

B 

83  Cancri      .... 

6 

9  11  43 

4-       3.4 

4-  18  15  17 

—      15.1 

126 

C 

0    Argus      .... 

1 

9  11  46 

4-       0.8 

—  69  10  57 

—      14.8 

127 

A    B           C 

t     Argus       .... 

2 

9  13  37 

4-      1.6 

—  58  43  47 

—      14.9 

128 

K    Argus       .... 

4 

9  18    5 

4-      1.9 

—  54  27  26 

—      15.3 

129 

A 

1    Draconis  .... 

4.  4 

9  18  20 

4-      9.2 

+  81  53  50 

—      15.3 

130 

A    B    J    C 

o    Hydra?      .... 

2.  3..  .2.  7 

9  21  12 

4-3.0 

—    8     5  47 

—      15.4 

THE    VERTICAL    OF    THE    POLE    STAR. 
GENERAL  STAK  CATALOGUE— Continued. 


45 


No. 

A.   B.    J.   C. 

Name. 

Mag. 

a  1870.0. 

Annual 
•ariation. 

J  1870.0. 

Annual 
variation. 

h.  m.  s. 

«. 

o     /     // 

„ 

131 

A 

24  (d)  Ursae  Majoris    . 

4.6 

9  22  56 

+       5.4 

+  70  23  57 

—      15.5 

132 

A    B    J 

6    Ursae  Majoris    .     . 

3 

9  24    9 

4-       4.1 

4-  52  16    4 

—      16.  2 

133 

A    B 

£     Leonia      .... 

3 

9  38  28 

\-      3.4 

4-  24  22  17  , 

—      16.4 

134 

•u    Argus      .... 

3 

9  43  51 

+       1.5 

—  64  28  10  | 

—      16.6 

135 

A                 C 

ft    Leouia      .... 

4 

9  45  22 

+       3.4 

4-  26  37    4 

—      16.7 

136 

B 

TT   Leonia      .... 

5 

9  53  21 

+       3.2 

4-     8  40    0 

—      17.1 

137 

A    B    J    C 

a    Leonis 

1.4 

10    1  27 

4-       3.2 

+  12  36    6 

—      17.  4 

138 

A 

32  Ursa?  Majoris    . 

6 

10    8  34 

+       4.4 

-f  65  45  19 

—      17.8 

139 

£    Leouis      .... 

3 

10    9  28 

4-       3.4 

4-24    3  52 

—      17.7 

140 

A    B    J    C 

y1  Leonis     .... 

2 

10  12  48 

4-       3.3 

4-  20  29  53 

—      18.0 

141 

/x    Ursa?  Majoris    . 

3 

10  14  35 

+       3.6 

f  42    9  10 

—      17.9 

142 

A 

9    Draconis  .... 

4.6 

10  23  58 

+       5.3 

4-  76  22  52 

—      18.4 

143 

A    B           C 

p    Leonis      .... 

4 

10  25  58 

4-       3.2 

-f     9  58  28 

-      18.4 

144 

0    Argus      .... 

3 

10  38  19 

+      2.1 

—  63  42  4H 

—      18.8 

145 

A    B          C 

7]    Argus      .... 

1  6 

10  40     1 

+       2.3 

—  59     0     3 

—      18.8 

146 

It    Argus      .... 

3 

10  41  12 

4-       2.6 

—  48  44    0 

—      18.9 

147 

A    B 

I     Leonis      .... 

5 

10  42  25 

+      3.2 

4-  11  13  56 

—      18.9 

148 

C 

v    Hydrae      .    .. 

3.4 

10  43    3 

+       2.9 

—  15  30  52 

—      18.7 

149 

C 

(3    Urae  Majoris    .     . 

2.4 

10  53  59 

+       3.7 

+57     4  43 

—       19.2 

150 

A    B    J    C 

a    Ursa3  Majoris    .     . 

2 

10  55  41 

+      3.8 

+  62  27     7 

—      19.4 

151 

B    J 

X    Leonis      .... 

5 

10  58  19 

+       3.1 

+     8    2  17 

—      19.4 

152 

rl  Ursee  Majoris     . 

3 

11    2  21 

+       3.4 

4-  45  12  10 

—      19.5 

153 

A    B    J 

o    Leouis      .... 

2.4 

11     7  12 

+       3.2 

4-  21  14    8 

—      19.6 

154 

0    Leouis      .... 

3.4 

11    7  2} 

+       3.1 

4-  16    8  24 

—      19.5 

155 

A    B    J 

J    Crateris    .... 

3.4 

11  12  51 

+       3.0 

—  14     4  32 

—      19.4 

156 

A 

T    Leonis      .... 

5 

11  21  15 

+      3.1 

4-     3  34  19 

—      19.8 

157 

A                 C 

A    Draconis  .... 

34 

11  23  39 

4-       3.6 

+  70    2  52 

—    .19.9 

158 

A    B 

v    (91)  Leouis    .     .  •   . 

4.6 

11  30  18 

+       3.1 

—    0    6  22 

—      19.8 

159 

A    B    J    C 

13    Leonis      .... 

2 

11  42  26 

+       3.1 

+  15  17  56 

—      20.1 

160 

J    C 

ft    Virginia  .... 

3.5 

11  43  55 

4-      3.1 

4-     2  29  50 

—      20.3 

161 

A    B    J    C 

y    Ursa)  Majoris    . 

2.4 

11  46  59 

4-       3.2 

+  54  25    3 

—      20.0 

162 

A 

o    Virginia  .... 

4 

11%  58  35 

4-      3.1 

4-     9  27  18 

—      20.0 

163 

B 

s     Corvi  

3 

12    3  26 

+       3.1 

—  21  53  48 

—      20.  0 

164 

A 

4    Draconis  .... 

4.6 

12    6    5 

+      2.9 

4-  78  20  18 

—      20.1 

165 

6    Crucis      .... 

3 

12    8  15 

4-       3.1 

—  58    1  29 

—      20.0 

166 

C 

6    Ursae  Majoris    . 

3.4 

12    8  59 

4-      3.0 

-f  57  45  16 

—      20.  1 

167 

Y    Corvi  

2 

12    9    7 

4-       3.1 

—  16  49  10 

—      20.0 

168 

A    B 

(1    Chamreleontis   .     . 

5 

12  10  46 

+      3.3 

—  78  35  26 

—      20.0 

169 

A    B          C 

TJ    Virginia  .... 

3.4 

12  13  15 

4-      3.1 

4-     0    3  21 

—      20.0 

170 

A     B          C 

a*  Crucis       .... 

1 

12  19  23 

+       3.3 

—  62  .2  38 
/v 

—      19.9 

171 

C 

/   Corvi  

2.4 

12  23    9 

4-       3.  I 

—  16  47  28 

—      20.  1 

172 

y    Crucis      .... 

2 

12  23  58 

+       3.3 

—  56  23    0 

—      20.1 

173 

A    B 

/3   Corvi  

2.4 

12  27  34 

+       3.1 

—  22  40  40 

—      20.  0 

174 

A 

K    Dracouis       .     . 

3.4 

12  27  55 

+      2.6 

4-  70  30  17 

—      19.  9 

175 

y    Centauri 

3 

12  34  21 

4-       3.3 

—  48  14  45 

—      19.  9 

176 

B              C 

y1  Virginia  .... 

2.6 

12  35    4 

+       3.0 

—    0  44  12 

—      19.8 

177 

J        I 

Y*  Virginia  .... 

2.6 

12  35    5 

4-       3.0 

—    0  44    6 

—      19.8 

178 

C 

p    Crucis      .... 

2 

12  40    8 

+       3.4 

—  58  58  34 

—      19.7 

179 

A 

32  Camelopardalis 

4.6 

12  48  12 

+       0.4 

+  84     7    9 

—      19.6 

180 

C 

e    LTrsa3  Majoris    .     . 

2 

12  48  18 

+       2.7 

4-  56  39  56 

—      19.7 

181 

1 

J    Virginia   .... 

3 

12  49    4 

+      3.0 

4-     4     6  15 

—      19.7 

182 

A    B    J 

12  Canum  Venat.  .     . 

3 

12  49  57 

+       2.8 

4-39     1  16 

—      19.5 

183 

C 

e     Virgin  is    .           .     . 

2.6 

12  55  43 

+       3.0 

4-  11  39  32 

—      19.5 

184 

A    B 

0    Virginia  .... 

4.4 

13    3  13 

+       3.1 

—    4  50  40 

—      19.3 

185 

y    Hydra      .... 

3 

13  11  51 

4-      3.2 

—  22  29    2 

—      19.  1 

186 

t    Centauri  .... 

3 

13  13  18 

+       3.4 

—  36     1  34 

—      19.1 

187 

A    B    J    C 

«    Virginia        .     .     . 

13  18  21 

4-       3.2 

—  10  28  55 

—      18.9 

188 

£    Uraa>  Majoris    . 

2 

13  18  41 

4-       2.4 

4-  55  36  17 

—      18.9 

189 

A    B    J    C 

?    Virginia  .... 

3.4 

13  28    4 

+       3.1 

+     0    4  11 

—      18.5 

190 

£     Centauri  .... 

3 

13  31  40 

+       3.7 

—  52  48  15 

—      18.6 

191 

A    B    J    C 

»7    Ursa1  Majoria    .     . 

2 

13  42  25 

4-      2.4 

+  49  57  47 

—      18.1 

192 

£    Centauri 

3 

13  47  26 

4-       3.7 

—  46  38  50 

—      18.0 

193 

A    B    J 

1    Bootis       .     .     . 

3 

13  48  30 

4-       2.9 

4-  19    3    1 

—      18.2 

194 

A    B          C 

P    Centanri  .... 

1 

13  54  40 

4-       4.  2 

—  59  44  40 

—       17.7 

195 

B 

T    Virginia  .... 

4 

13  55    2 

+       3.  0 

4-     2  10  28 

—      17.6 

46 


THE    PORTABLE    TRANSIT    INSTRUMENT    IN 
GENERAL  STAK  CATALOGUE— Continued. 


Xo. 

A.   B.   J.   C. 

Name. 

Mag. 

a  1870.0. 

Annual 
variation 

J  1870.0. 

Annual 
variation. 

h.  m.  s. 

s. 

o     /     // 

„ 

196 

c 

6    Centauri  .     .     .  '  . 

3 

13  59    3 

+       3.5 

—  35  43  48  |—      18.  1 

197 

A                 C 

a    Draconis  .... 

3.4 

14    0  52 

+       1.6 

4-  64  59  50   —      17.  4 

198 

A    B    J    C 

a    Bootis       .... 

1 

14    9  44 

4r       2.7 

4-  19  51  38   —      18.9 

199 

A 

jp.  Bootis       .... 

3.6 

14  20  46 

4-       2.0 

4-  52  27    9   —      16.  8 

200 

B 

p    Bootis       .... 

3.6 

14  26  14 

4-      2.6 

4-  30  56  3<\  —      16.0 

201 

y    Bootis      .... 

2.6 

14  26  51 

4-       2.4 

4-  38  52  40  !—      16.  0 

202 

.77    Centauri  .... 

3 

14  27  15 

4-       3.8 

—  41  35    9  j—      16.  2 

203 

A 

5    Ursre  Minoris    . 

4.7 

14  27  50 

—      0.2 

4-  76  16  25   —      16.  0 

204 

A     B          C 

a2  Centauri  .... 

1 

14  30  48 

4-       4.0 

—  60  17  39    —      15.  0 

205 

a    Lupi    

3 

14  33  16 

4-      3.9 

—  46  49  43 

—      15.9 

206 

C       £    Bootis       .... 

3.4 

14  34  57 

4-       2.9 

4-  14  17  14 

—      15.7 

207 

A    B                  £     Bootis       .... 

2.4 

14  39  19 

4-      2.6 

+  27  37  24 

—      15.4 

208 

J          i  a1  Libra}       .... 

6 

14  43  27 

4-       3.3 

—  15  27    2 

—       15.2 

209 

A    B    J    C 

a2  Libra}       .... 

2.4 

14  43  41 

+      3.3 

—  15  29  59 

—      15.2 

210 

/?    Lupi    

3 

14  50     2 

4-       3.9 

—  42  36  31 

—      15.0 

211 

K    Centauri  .... 

3 

14  50  43 

4-       3.9 

—  41  24  56 

—       14.8 

212 

A    B    J    C 

(i    Ursa?  Minoris    . 

2 

14  51     7 

—      0.3 

+  74  41  11 

—      14.8 

213 

A                 C 

li    Bootis       .... 

3 

14  57     3 

4-      2.3 

4-  40  54  15 

—       14.4 

214 

B    J 

ip   Bootis      .     .     .     . 

4.4 

14  58  53 

4-      2.6 

+  27  27  22 

—       14.2 

215 

y    Trianguli  Australis 

3 

15     6  49 

4-      5  5 

—  68  11  47 

—      13.9 

216 

A    B 

0    Libra}       .... 

o 

15  10     1 

+       3.2 

—    8  54    5 

—      13.6 

217 

J    Bootis      .... 

3 

15  10  16 

4-      2.4 

4-  33  48    6   —      13.  7 

218 

A 

/*i   Bootis       .... 

3.6 

15  19  35 

4-       2.3 

4-  37  50    4   —      12.  8 

219 

A                 C 

\2  Ursa,1  Minoris    .     . 

3 

15  20  57 

—      0.1 

-   72  17  48    —       12.  8 

220 

t     Draconis        .     . 

3 

15  22    3 

4-       1.3 

+  59  25  2'J 

—      12.8 

221 
222 

y    Lupi    
J    Serpentis 

3 
3.4 

15  26  29 
15  28  36 

+       4.0 

4-       2.9 

—  40  43  44 

4-  10  58  34 

—      12.7 
—      12.3 

223 

A    B    J    C 

a    Corona}  Borealis     . 

2 

15  29  11 

4-      2.5 

4-27    9  15  j—      12.  3 

224 

A    B    J    C 

a     Serpentis       ...            2.  4 

15  37  52 

4-       3.0 

4-     6  50  12  1—      11.  6 

225 

0    Trianguli  Australia 

3 

15  43  43 

4-      5.2 

—63     1  30 

—      11.7 

226 

A                C 

£    Serpentis       .     . 

3.4 

15  44  20 

4-      3.0 

4-     4  52  15 

—      11.1 

227 

A    B    J 

§    Ursa}  Minoris     .      . 

4.4 

15  48  45 

—      2.3 

4-  78  11  35   —      10.  9 

228 

y    Serpentis 

3.6 

15  50  26 

4-       2.8 

4-16    5  19   —       12.  0 

229 

IT    Scorpii      .... 

3 

15  51     0 

4-       3.6 

—  25  44  17   —      10.  8 

230 

A 

£    Corona}  Borealis     . 

4 

15  52  12 

+      2.5 

4-  27  15  21 

—      10.6 

231 

A 

S    Scorpii      .... 

2.4 

15  52  39 

4-       3.5 

—  22  14  57 

—      10.6 

232 

A    B 

0l  Scorpii           .     .     . 

2 

15  57  53 

-       3.5 

—  19  26  50 

—      10.2 

233 

0    Draconis 

3.6 

15  59  28 

4-       1.1 

4-  58  54  46 

—        9.8 

234 

235 

A 

A    B 

2320  Groombriclge     .     . 
6    Ophiuchi       .     .     . 

5.6 
3 

16    5  58 
16     7  32 

4-      0.1 
4-       3.1 

4-68    9  10 
—    3  21  27 

—        9.5 
—        9.6 

236 

j  £     Ophiuchi       .     .     . 

3.4 

16  11  26 

4-       3.2 

—    4  22  25 

-        9.2 

237 

A                         T    Herculis  .... 

3.4 

16  15  50 

-f       1.8 

4-  46  37  27 

—        8.8 

238 

y    Herculis  .... 

3 

16  16  11 

4-      2.6 

4-  19  28    6 

—        8.8 

239 

A    B    J    C 

a    Scovpii     .... 

1.4 

16  21  26 

4-       3.7 

—26    8  26 

—        8.4 

240 

A    B 

r)    Draconis  .... 

2.6 

16  22  15 

4-      0.8 

4-  61  48  33 

—        8.2 

241 

0    Herculis       .     .     . 

2.4 

16  24  38 

4-      2.6 

4-  21  46  21 

—        8.2 

242 

A 

15  (A)  Draconis      .     . 

5 

16  28  15 

—      0.1 

4-  69    2  58 

—        7.8 

243 

A 

£t  Ophiuchi       .     ,     . 

2.6 

16  30    0 

4-       3.3 

—  10  18    5 

—        7.6 

244 

A    B          C 

a    Trianguli  Australia 

2 

16  34  56 

6.3 

—  68  47    4 

—        7.4 

245 

B    J 

g    Hercinis        .     .     . 

2.6 

16  36  23 

4-       2.3 

-  31  50  24 

—        6.8 

246 

A 

»7    Herculis 

3 

16  38  26 

4-       2.1 

-L  39  10  16 

-J        7.0 

247 

c 

£    Scorpii      .... 

3 

16  41  45 

+       3.9 

—  34    3  17 

—        7.  1 

248 

Hl   Scorpii     .... 

3 

16  42    4 

4-      4.0 

—  37  49  21 

6.8 

249 

S*2  Scorpii     .... 

3 

16  45  26 

4-       4.2 

—42    8  14 

-        6.9 

250 

A    B    J 

K    Ophiuchij      .     .     . 

3.4 

16  51  31 

4-       2.8 

4-     9  34  45 

—        5.9 

251 

C    j  £    Herculis  .... 

3.4 

16  55  19 

4-      2.3 

4-  31     7  12 

—        5.6 

252 

A                      j  d    Herculis  .... 

5 

16  56  48 

4-      2.2 

4-  33  45  30 

—        5.4 

253 

A     B                   £     Ursa}  Minoris    .     . 

4.4 

16  59  23 

-      6.4 

4-  82  14  49 

—        5.2 

254 

n    Ophiuchi       .     .     . 

2.4 

17    2  53 

4-       4.3 

-  15  33  40 

—        5.4 

255 

£    Draconis  .... 

3 

17    8  25 

4-      0.2 

4-  65  52  29 

—        4.5 

256 

A    B    J    C       a1   Herculis  .... 

3.1.  ..3.  9 

17     8  43 

4-      2.7 

4-  14  32  27 

—        4.4 

257 

6    Herculis  .... 

3 

17     9  41 

4-      2.5 

-  24  59  38 

—        4.6 

258 

B                   &    Ophiuchi       .      .     . 

3.4 

17  14    2 

3.7 

-  24  51  59 

—        4.0 

259 

y    Ara}  • 

3 

17  14  28 

4-      5.0 

—  56  15    4 

—        4.1 

260 

/?    Ar«     3             17  14  30 

4-      5.0 

-  55  24  13 

—        4.2 

THE    VERTICAL    OF    THE    POLE    STAR. 
GENERAL  STAR  CATALOGUE— Continued. 


47 


Xo. 

A.   B.   J.   C. 

Name. 

Mag. 

a  1870.0. 

Annual 
variation. 

(5  1870.0. 

Annual 
variation. 

h.  m.  s. 

g 

0         /    II 

n 

2G1 

A 

b    (44)  Ophiuchi      .     . 

5 

17  18  26 

+       3.7 

—24     3  11 

—        3.7 

•26-2 

a    Aric     

3 

17  21  48 

r        4.6 

—  49  46  10 

—        3.6 

263 

A    Scorpii     .... 

3 

17  24  47 

r       4.  1 

—  37    0  19 

—        3.1 

264 

A    B    J 

ft    Dracoiiis  .... 

2.6 

17  27  30 

-;-     i.  4 

4-  52  23  55 

—        2.8 

263 

0    Scorpii      .... 

3 

17  27  56 

+       4.3 

—  42  54  47 

—        3.1 

266 

A    B    J    C 

a    Opliiuchi  .... 

2 

17  28  54 

4-       2.8 

4-  12  39  54 

—        2.  9 

267 

*    Scorpii      .... 

3 

17  33  30 

+       4.1 

—  38  57  38 

—        2.5 

268 

C 

0   Opliiuchi  .... 

3 

17  37     3 

4-       3.0 

+     4  37  25 

—        1.9 

26» 

A 

oj  Draconia  .... 

5 

17  37  43 

—      0,4 

4-  68  49    2 

—        1.  6 

270 

A    B    J 

H    Herculis  .... 

3.4 

17  41  22 

+       2.3 

+  27  47  54 

2.  4 

271 

A 

i//1  Herculis,  (pr.)  .     . 

4.  4 

17  44  15 

—       1.1 

-f  72  12  43 

—        1.6 

272 

A    B    J    C 

y1  Dracoiiis  .... 

2  4 

17  53  35 

4-      1.4 

+  51  30  18 

—        0.6 

273 

A 

y2  Sagittarii 

a  4 

17  57  27 

-t-      3.9 

—  30  25  23 

—        0.4 

274 

A    B 

Hl   Sagittarii      .     .     . 

4 

18    5  59 

+       3.6 

—  21     5  25 

4-        0.5 

275 

A    B 

<r    Octantia  .... 

6 

18    6  19 

+  109.2 

—  89  16  43 

4-        0.  6 

276 

A    B    J    C 

6    Ursa;  Miuoris    . 

4.4 

18  14  16 

—    19.4 

4-  86  36  20 

4-        1.3 

2*77 

278 

A 

77    Serpentis 
e    Sagittarii      .     .     . 

3 

2.6 

18  14  35 
18  14  52 

+       3.1 

+       4.0 

—    2  55  49 
—  34  26  33 

4-        0.6 
4-        1.2 

279 

A    Sagittarii      .     .     . 

3 

18  19  57 

4-      3.7 

—  25  29  30 

4-         1.4 

280 

A 

1     Aquilie     .... 

4.4 

18  28    8 

+       3.3 

—    8  19  58 

4-        2.2 

281 

A    B    J     C 

a    Lyric  

1 

18  32  32 

4-      8.0 

+  38  39  51 

4-        3.  1 

282 

A    B    J    C 

0l  Lyne  

3.  5...  4.  5 

18  45  17 

+      2.2 

+  33  12  47 

4-        3.9 

283 
284 

A 

A 

o    Sagittarii 
50  Dracoiiis  .... 

o  4 
6 

18  47  12 
18  50  33 

+       3.7 
—      1.9 

—  26  27  19 
4-  75  16  44 

+        4.0 
4.4 

285 

y    Lyric  

3.4 

18  54    5 

+       2.2 

4-  32  30  48 

+         4.6 

286 

C 

g    Sagittarii      .     .     . 

3.4 

18  54  20 

+     .  3.  8 

—30    3  50 

4-        4.6 

287 
288 

A    B 

A    Aquilse     .... 
g    Aquila>     .... 

3.4 
3 

18  59  21 
18  59  26 

+       3.2 

+       2.8 

—    5    4  30 
+   13  40  21 

J-         5.0 
4-        5.1 

289 

TT    Sagittarii      .     .     . 

3 

19    2    2 

+       3.6 

—  21  13  38 

4-        5.3 

290 

A 

d    Sagittarii       .     .     . 

5 

19  1U    2 

+      3.5 

—  19  10  55 

4-        6.1 

291 

B 

a)    Aquilas     .... 

5.6 

19  11  43 

+       2.8 

+  11  21  46 

4-        6.2 

292 

A                 C 

S    Dracoiiis       .     .     . 

3 

19  12  31 

0.0 

4-  67  25  58 

4-        6.3 

293 

A 

T    Dracoiiis 

5 

19  18    2 

—      1.1 

4-  73    6  47 

4-        6.8 

294 

A     B    J 

S    Aquilie. 

3.4 

19  18  57 

4-       3.0 

+     2  51  23 

4-        6.  H 

295 

C 

0    Cygni       .... 

3 

19  25  29 

+      2.4 

-f-  27  41  20 

4-         7.  J 

296 

B 

ft2  Sagittarii      .     .     . 

4.6 

19  28  47 

4-      3.7 

—  25  10    3 

4-        7.6 

297 

A 

K    Aquilas     .... 

5 

19  29  54 

+       3.  2 

—    7  18  51 

4-        7.7 

298 

A    B    J    C 

y    Aquilae     .... 

3 

19  40    5 

+       2.9 

+  10  17  55 

+        8.5 

299 

3    Cvgui 

3 

19  40  55 

+       1  9 

4-  44  48  50 

1           HI 

300 

A    B    J    C 

~:J  o  .,  
a    Aquiloe     .... 

1.4 

19  44  26 

A.  J 

+      2.9 

+     8  31  37 

-t"             C.  O 

4-        9.2 

301 

>7     AquilsE     .... 

3.  5..  .4.  7 

19  45  51 

+      3.1 

+     0  40  26 

+        8.8 

302 

A 

e     Dracoiiis       .     . 

4 

19  48  36 

—      0.2 

+  69  56  11 

4-        9.2 

303 
304 

A    B    J    C 

A    B 

0    Aquila3     .... 
A    TJrsaj  Minoris    . 

4 
6.4 

19  48  56 
19  54  17 

+       2.9 
—    59.2 

+   .6    5    2 

-f  88  55    4 

4-        8.7 
9.  6 

305 

A 

T    Aquilae     .... 

5.6 

19  57  47 

4-      2.9 

4-     6  54  46 

4-        9.9 

306 

C 

0    Aquilie     .... 

3 

20     4  36 

+      3.1 

—    1  12  17 

4-      10.3 

307 
308 
309 

J 

A    B    J    C 
A 

a1  Capricorni    .     .     . 
a2  Capricorni    . 
K    Cephei      .... 

4.4 
3.4 
4.4 

20  10  26 
20  10  50 
20  13  13 

+       3.3 
-j-       3.3 
—      1.9 

—  12  54  28 
—  12  56  44 
+  77  19    0 

4-      10.8 
4-      10.8 
4-       11.0 

310 

0    Capricorni    .     .     . 

3 

20  13  42 

+       3.4 

—  15  11  22 

4-      11.  0 

311 

A    B          C 

a    Pavonis    .... 

2 

20  15  21 

+       4.8 

—57    8  54 

4-      11.  1 

312 
313 

C 
A 

y    Cygui     .... 
v    Capricorni    .     .     . 

2.6 
5 

20  17  34 
20  19  53 

+      2.2 
+       3.4 

+  39  50  31 
—  18  38    8 

f      11.  3 
4-      11.5 

314 

315 

B 
A 

p    Capricorni    .     .     . 
e    Delphini  .... 

5 
4 

20  21  26 

20  27    0 

+       3.4 

4-       2.9 

—  18  14  28 
4-  10  51  48 

4-      11.6 
4-      12.  0 

316 

a    Incli     .... 

3 

20  28  25 

4-       4.  3 

—  47  44  31 

—       1  1.  9 

317 
318 
319 

A 
C 

3241  Groombridge    . 
13    Pavonis    . 
a    Delphini  .... 

6.4 
3 
3.6 

20  30  33 
20  33  13 
20  33  36 

—      0.2 
+       5.5 

+       2.  8    • 

4-  72    5  28 
—  66  39  58 
4-  15  27  19 

4-       12!  2 
4-       12.  3 
12.4 

320 

A    B    J    C 

a    Cygni  

1.6 

20  37    0 

+       2.0 

4-  44  49    0 

4-       12.7 

321 

£     Cygni       .... 

2.6 

20  40  57 

4-      2.4 

4-  33  29    5 

-      13.  2 

322 
323 

A 
B 

H    Aquarii    .... 
32  Vulpeculaj    . 

4.6 
5.4 

20  45  38 
20  49    1 

+       3.2 
+       2.  6 

—    9  28    9 
4-  27  33  52 

4-       13.  2 
4-      13.5 

324 
325 

A 
A 

v    Cygni  
1879  Twelve  year  C. 

4 
6 

20  52  20 
20  53  24 

+      2.2 
—      2.5 

4-  40  40    4 

4-80    3  47 

4-       ]3.  7 

4-       13.  7 

48 


THE    PORTABLE    TRANSIT    INSTRUMENT,    ETC. 
GENERAL  STAR  CATALOGUE— Continued. 


No. 

A.  B.   J.   C. 

Name. 

Mag. 

a  1870.0. 

;  Annual 
variation. 

6  1870.0. 

Annual 
variation. 

h.    in.  s. 

s. 

o       /  // 

i, 

326 

A    B    J 

61lCvgni       .... 

5.4 

21     1     4 

+      2.7 

4-38    6  41 

4-       17.5 

327 

A    B 

£    Cygni  

3 

21     7  24 

+       2.6 

4-  29  41  42 

+       14.6 

328 

A    B    J    C 

a    Cephei      .... 

2.6 

21  15  28 

+       1.4 

4-  62    2    6 

4-       15.1 

329 

y    Pavouis    .... 

3 

21  15  40 

4-       5.1 

—  65  57  11 

4-      15.7 

330 

A 

1    Pegasi      .... 

4.4 

21  16    4 

+      2.8 

4-  19,15    0 

+       15.2 

331 
332 

A    B          C 
A    B    J 

ft    Aquarii    .... 
ft    Cephei      .... 

3 
3 

21  24  43 

21  26  58 

+       3.2 

+      0.8 

—    6    8  30 

4-  69  59  24 

4-       15.6 
4       15.7 

333 

A 

£    Aquarii    .... 

4.6 

21  30  50 

+       3.2 

—    8  26    9 

4-      15.9 

334 

A    B 

e    Pegasi      .... 

2.4 

21  37  48 

4-      2.9 

-f     9  16  49 

+       16.3 

335 

C 

J    Capricorn!    .     .     . 

3 

21  39  51 

4-       3.3 

—  16  42  54 

+       16.1 

336 

A 

11  Cephei     .... 

5 

21  40    0 

+       0.9 

4  70  42  46 

+      16.5 

337 

Y    Gruis  

3 

21  46    3 

4-       3.7 

—  37  58  33 

4       16.5 

338 

A 

\L    Capricorui    .     . 

5 

21  46  12 

+      3.3 

—14     9  45 

4       16.8 

339 

B 

16  Pegasi      .... 

5.4 

21  47    9 

+       2.7 

4-  25  18  52 

4-      16.8 

340 

A 

79  Draconis  .... 

6.4 

21  51  15 

4-       0.7 

4  73    5  14 

+      17.0 

341 

A    B    J    C 

a    Aquarii    .... 

3 

21  59    6 

4-       3.1 

—    0  57    1 

+       17.3 

342 

A    B          C 

a    Gruis  

2 

22    0    2 

4-       3.8 

—  47  35  20 

4-      17.2 

343 

C 

g    Cephei      .... 

3.6 

22     6  21 

+       2.1 

+  57  33  39 

4-      17.6 

344 

a    Tucanae    .... 

3 

22     9  34 

+       4.2 

—  60  54  20 

4-       17.7 

345 

A    B 

0    Aquarii    .... 

4.4 

22    9  58 

+       3.2 

—    8  25  46 

4-       17.8 

346 

C 

y    Aquarii    .... 

3.6 

22  14  57 

+       3.1 

—    2    2  28 

-1-       18.  0 

347 

A 

TT    Aquarii    .... 

4.6 

22  18  38 

+       3.1 

4-     0  43    6 

4-      18.1 

348 

A    B 

TI    Aquarii    .... 

3.6 

22  28  40 

+       3.1 

—    0  47  12 

4-       18.  4 

349 

A 

2-26  Cephei      .... 

5.4 

22  29  59 

4       1.1 

4-  75  33  23 

+       18.5 

350 

0    Gruis  

3 

22  34  54 

-f       3.  6 

—  47  33  40 

+       18.6 

351 

A    B 

S    Pegasi      .... 

3.4 

22  34  58 

+       3.0 

+10    9  14 

4-       18.7 

352 

T]    Pegasi      .... 

3 

22  36  55 

4-       2.8 

4-  29  32  32 

+       18.7 

353 

A 

i     Cephei      .... 

3.6 

22  45    3 

+       2.1 

-r  05  31     1 

+       18.8 

354 

A 

X    Aquarii    .... 

4 

22  45  50 

f       3.  1 

—     8  16  13 

+       19.  1 

355 

<5    Aquarii    .... 

3 

22  47  45 

4       3.2 

—  16  30  40 

4-       19.1 

356 

A    B    J    C 

a    Piscis  Australia 

1.4 

22  50  28 

+       3.3 

—  30  18  38 

4      19.0 

357 

0    Pegasi      .... 

2.  2...2.  7 

22  57  28 

+       2.9 

4-  27  22  43 

+       19.5 

358 

A    B    J    C 

o    Pegasi      .... 

0 

22  58  17 

4-       3.0 

4-  14  30  24 

4-       19.3 

359 

B    J    C 

y    Piscium   .... 

4 

23  10  26 

+       3.1 

-f     2  34  20 

4-       19.6 

360 

A 

o     Cephei     .... 

5.6 

23  13  18 

+       2.4 

+  67  24     0 

+       19.  6 

361 

B 

K    Piscium   .... 

4.6 

23  20  16 

4-      3.1 

+     0  32  39 

+       19.6 

362 

A 

6    Piscium   .... 

4.4 

23  21  22 

4-       3.0 

+     5  39  54 

+       19.7 

363 

A    B    J 

i     Piscium    .... 

4.4 

23  33  16 

+      3.1 

4-     4  55  18 

+       19.  5 

364 

A    B 

y    Cephei     .... 

3.4 

23  34    2 

+       2.4 

4-  76  54  25 

+       20.1 

365 

B 

o    Sculptoris     .     .     . 

4.4 

23  42    9 

+       3.1 

-  28  50  55 

+       19.9 

366 
367 

A 
A    B    J 

4163Groombridge     .     . 
w    Piscium   .... 

7 
4 

23  48  32 
23  52  38 

4-       2.8 
4-      3.1 

+  73  41  12 
-r     6    8  36 

+       20.  0 
+       19.9 

368 

c 

2    Ceti     . 

4.4 

23  57    5 

+       3.1 

—18    3  34 

4-      20.  1 

14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 

8    Oct'58  ? 


LD  21-50m-8,'57 
(,C8481slO)476 


General  Library 

University  of  California 

Berkeley 


^^ 


PAMPHIET   BINDER 

Syracuse,  N.  Y. 
Stockton.  Colif. 


